PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

MA6453 PROBABILITY & QUEUEING THEORY

UNIT I

RANDOM VARIABLES AND STANDARD DISTRIBUTIONS

SYLLABUS: Discrete and continuous random variables – Moments – Moment generating functions
– Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions.

COURSE OBJECTIVE: Know the characteristics of Probability distributions by identifying the

discrete and continuous random variables. The Binomial distribution is used in quality control of
item manufactured by a production line when each item is classified as either defective or non-
defective. The Poisson distribution applies in its own right where the possible number of discrete
occurrences is much larger than the average number of occurrences in a given interval of time or
space. It is also used as a convenient approximation to the binomial distribution in some
circumstances

PART-A

1. If a random variable X takes the values 1,2,3,4 such that P(X=1)=3P(X=2)=P(X=3)=5P(X=4).

Find the probability distribution of X. (Nov. Dec.2012)
Solution:
  
Assume P(X=3) = α. By the given equation, P ( X  1)  P ( X  2)  P ( X  4)  .
2 3 5
For a probability distribution (and mass function)  P(x) 1
P(1)+P(2)+P(3)+P(4) =1
   61 30
    1   1 
2 3 5 30 61
15 10 30 6
P ( X  1)  ; P ( X  2)  ; P ( X  3)  ; P ( X  4) 
61 61 61 61

The probability distribution is given by

X 1 2 3 4
15 10 30 6
p( x)
61 61 61 61

2. Let X be a continuous random variable having the probability density

 2
function , x  1 Find the distribution function of x.
f ( x)   x3
 0 , otherwise
Solution:
x
x x
2  1  1
F ( x)   f ( x) dx   dx     1
1 1x
3  x 
2
x
2
1

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3. A random variable X has the probability density function f(x) given by

 x
f ( x)  cx e , x  0 . Find the value of c and CDF of X.
 0, otherwise
Solution:
 

 f ( x ) dx  1   cx e  x dx  1  c  x e  x  e  x  
0  1  c 1  1  c  1
0 0

 
x x x
F x    f ( x) dx   cx e dx   x e  x dx   x e  x  e  x
x x
0 1  x ex  ex
0 0 0
4. A continuous random variable X has the probability density function f(x) given
x
by f ( x)  ce ,  x   . Find the value of c and CDF of X.
Solution:

  

 f ( x) dx  1   c e dx 1  2 c e
x x
dx 1
  0

 2 c e  x dx 1  2c  e  x  

0 1  2c1 1  c 
1
2
0

Case (i ) x  0

 
x x x
F x  
1 x
 f ( x ) dx   c e dx  c  e x dx  c e x
x x
  e
  
2
Case ( ii ) x  0
x x  x x 0 0 x
F x   x dx  c e  x dx  c  e x   e  x 
 f ( x ) dx  ce  e 
dx  c
   
 c
  0
   0
 c  ce  x  c  c  2  e  x    x
1
2  e 
  2 
1 x
 2 e , x0
F ( x)   x 
1
  2  e , x  0
 2  

  2 x
5. If a random variable has the probability density f ( x)  2 e , x  0 . Find the
 0, otherwise
probability that it will take on a value between 1 and 3. Also, find the probability that it will
take on value greater than 0.5.

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 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

Solution:
3 3 2x 3
 2x  2 6
P(1  X  3)   f ( x) dx   2e dx   e   e e
1 1  1
  2x 
 2x  1
P( X  0.5)   f ( x) dx   2e dx   e  e
0.5 0.5   0.5

6. Is the function defined as follows a density function?

 0, x2
1
f ( x)   3  2 x , 2 x4
18
 0, x4
Solution:
4
4 4  3  2 x 2 
 f ( x ) dx 
1
 18 3  2 x  dx    1
2 2 
 72  2
Hence it is density function.

7. The cumulative distribution function (CDF) of a random variable X is

x
F ( X )  1  (1  x) e , x  0 . Find the probability density function of X.
Solution:
 
f ( x)  F   x   0  1  x  e  x  1 e  x     x e x , x0

8. The number of hardware failures of a computer system in a week of operations has the
following probability mass function:
No of failures : 0 1 2 3 4 5 6
Probability : 0.18 0.28 0.25 0.18 0.06 0.04 0.01
Find the mean of the number of failures in a week.
Solution:
E ( X )   x P ( x)  (0)(0.18)  (1)(0.28)  ( 2)(0.25)  (3)(0.18) 
( 4)(0.06)  (5)(0.04)  (6)(0.01)
 1.92
6 x(1  x), 0  x 1
9. Given the p.d.f of a continuous r.v X as follows f ( x)   .
 0, elsewhere
Find the CDF of X.
x x x x
2  2 3 2 3
Solution: F ( x)   f ( x) dx   6 x(1 _ x) dx   6 x _ 6 x dx  3x  2 x   3x  2 x
0 0 0  0

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10. A continuous random variable X has the probability function f ( x )  k (1  x ), 2  x  5 .

Find P(X<4).
Solution:
 1  x 2 
4 5 5

 k  1  x  dx  1  k 
27 2
 f ( x)dx  1
2 2  2 2
  1 k
2
 1 k 
27
4
4
2 4 2  1  x 2 
P( X  4)   f ( x) dx   1  x  dx     25  9 
1 16
27 27  2  25 27
2 2 2
12.5 x 1.25 0.1  x  0.5
11. Given the p.d.f of a continuous R.V X as follows f ( x)  
 0, elsewhere
Find P(0.2 < X < 0.3)
Solution:
0 .3
0 .3
 x2 
P (0.2  X  0.3)   (12.5 x  1.25) dx  12.5  1.25 x 
0 .2  2  0 .2

 1.25 50.3  0.3  50.2   0.2
2 2

 0.1875
12. If the MGF of a continuous R.V X is given by M X t   3 . Find the mean and variance of X.
3t
Solution:

1 2 3
3 1  t t t  t 
MX t      1   1       ...
3t 1
t  3 3  3  3
3
E( X )  coefficient of
1
3
t  1! 
is the 
mean, E( X 2 )  coefficient of t2 2!   1
9
2! 
2
9
2 1 1
Variance  E( X 2 ) E( X )2   
9 9 9
4
1 t
13. If the MGF of a discrete R.V X is given by M X t   1  2e  , find the distribution of X.
81  
Solution:

M X t  
1
81
 4
1  2e t   1
81
  
1  4C1 2e t  4C2 2e t   2
 
 4C3 2e t
3
 4C4 2e t  4

1 8 t 24 2t 32 3t 16 4t
  e  e  e  e
81 81 81 81 81
By the definition of MGF,
M X t  
tx t 2t 3t 4t
e p ( x)  p (0)  p (1)e  p (2)e  p (3)e  p ( 4) e

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On comparison with above expansion the probability distribution is

X 0 1 2 3 4
1 8 24 32 16
p( x)
81 81 81 81 81
1

14. Find the MGF of the R.V X whose p.d.f is f ( x )  10 , 0  x  10 . Hence find its mean.
 0, elsewhere
Solution:
10
1  e tx  1  e10t  1  1  
10
100t 2 1000t 3
M X t    e dx    
1 tx
  
 t  10t    1  10t    ....  1 
0
10 10  0
t 10    2! 3! 
1000 2
 1  5t  t  .....
31
Mean  coefficient of t  5
k
15. Given the probability density function f ( x)  ,   x   , find k and C.D.F.
1  x2
Solution:
 

 f ( x) dx  1 
k
 1 x 2
 

  
dx  1  k tan 1 x  1  k tan 1   tan 1    1 
 

   1
 k    1 k 
2 2 

     1 
  1
x
k 1 1
 f ( x) dx  
x
F ( x)  dx  tan 1 x   tan 1   tan 1  x    tan 1 x   cot 1 x
 
1 x 2
  2  

16. It has been claimed that in 60 % of all solar heat installation the utility bill is reduced by atleast
one-third. Accordingly what are the probabilities that the utility bill will be reduced by atleast
one-third in atleast four of five installations?
Solution:

Given n=5, p=60 % =0.6 and q=1-p=0.4

p( x  4)  p[ x  4]  p[ x  5]
 5c 4 (0.6) 4 (0.4) 5 4  5c5 (0.6) 5 (0.4) 55
 0.337

17. The no. of monthly breakdowns of a computer is a r.v. having poisson distribution with mean
1.8. Find the probability that this computer will function for a month with only one breakdown.
Solution:

e   x e 1.8 (1.8)1
p( X  x)  , given   1.8 , p( x  1)   0.2975
x! 1!

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 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

18. In a company 5 % defective components are produced. What is the probability that atleast 5
components are to be examined in order to get 3 defectives?
Solution:

To get 3 defectives, 3 or more components must be examined.

p=5 % =0.05, q = 1- p=0.95 and k=success=3

p ( X  x)  ( x  1)c k 1 p k q x  k , x  k , k  1, k  2,...
p ( x  5)  1  p ( x  5)
 1   p ( x  3)  p ( x  4)

1  2c 2 (0.05) 3 (0.95) 0  3c 2 (0.05) 3 (0.95)1 
1  0.00048  0.9995

19. A discrete R.V X has mgf . Find E(x), var(x), and p(x=0).
Solution: Given

We know that mgf of poisson is

Therefore λ = 2. In poisson E(x) = var(x) = λ

 Mean E ( x )  var ( x )  2

e  x
p ( X  x) 
x!

e   0
 p( X  0)   e   e 2  0.1353
0!

20. Find the mean and variance of geometric distribution.

Solution:

The pmf of Geometric distribution is given by

x 1
p ( X  x)  p q , x  1,2,3,.....
 
Mean E ( x)   x p( x)   x p q x 1  p  x q x 1  p 1 q11  2q 2 1  3q31  .....
 
x 1 x 1
 p 1 2q  3q 2  .....  p 1  q 2  p p 2  p 1 
1
  p

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 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

  
E ( x 2 )    x  x  1   x  p q x  1   x ( x  1) p q x  1   x p q x  1
x 1 x 1 x 1
 1(1  1) p q 1  1  2(2  1) p q 2  1  3(3  1) p q 3  1  .....  1
p
 
 2 p  2(3) p q  3(4) p q 2  .....  1  2 p 1  3 q  6 q 2  .....   1
p   p
 2 p 1  q   3  1  2 p p  3  1  2  1
p p p2 p

2
2 1 1
Variance  E ( x )  E ( x)  2      2   2
2 2 2 1 1
p p  p p p p
1 1 1 p q
 2
  2  2
p p p p

21. Find the MGF geometric distribution.

Solution: The PMF of geometric distribution is given by

 
Mgf M x (t )  E (e tx )   e tx p ( x)   e tx p q x1   e tx p q x q 1
x 1 x 1


p



q x1
  p
e t q x   qe t 
x

q x1
x p t
q

qe  qe t      qe 
2 t 3

 ...


p t
q
 2
qe 1  qe t  qe t  ...   

 pe t 1  qe t 
1
1  qe t
 1  qe t
1 sinh at
22. Show that for the uniform distribution f ( x )  ,  a  x  a , the mgf about origin is
2a at .
1
Solution: Given f ( x )  , a  x  a
2a

MGF
 a
1  e tx 
 
aa
1 1
  e f ( x)dx   e
2a a
M x (t )  E e tx tx
dx  e tx
tx
dx   
 a
2a 2a  t   a


1 at
2at

e  e at 
1
2at
2 sinh at
sinh at

at

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23. Define exponential density function and find mean and variance of the same.

Solution: The density function of exponential distribution is given by f ( x )   e  x , x  0

   
  xe  x e  x 
Mean  E x    x f ( x) dx   x e
 x
dx    x e  x
dx     2 
 0 0    0
  1   1 
   ( 0  0)   0  2     2 
     
1



   
  x 2 e  x 2 xe  x 2e  x 
  x
Ex 2 2
f ( x) dx   x  e
2  x
dx    x e 2  x
dx      3 
 0 0   2  0
  2   2 2
   ( 0  0  0)   0  0  3      3   2
      

 
Variance  E x  E ( x)2 2 2 1 2 1 1
 2    2  2  2
    

24. State memory less property of exponential distribution.

Solution:
If X is exponentially distributed with parameter  , then for any two positive integers ‘s’ and ‘t’
P( X  s  t / X  s)  P( X  t ) .

25. A continuous random variable X that can assume any value between x  2 & x  5 has a
density function given by f ( x )  k (1  x ) . Find P(X>4). (Nov/ Dec. 2012)

Solution: Since f(x) is a density function,

5
5
 x2   25 4  27  2
2 k (1  x ) dx  1  k 


x    k  5 
2 2  2
 2    k   1  k 
2  2  27

2  25 16  11
5
2
P ( X  4)   (1  x)dx  1   
4
27 27  2 2  27

26. Identify the random variable and name the distribution it follows, from following statement:
“A realtor claims that only 30% of the houses in a certain neighbourhood appraised at less than
Rupees 20 lakhs. A random sample of 10 houses from the neighbourhood is selected and
appraised to check the realtor’s claims acceptable are not”. (Nov/ Dec. 2012)

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 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

Solution:

X is a random variable that a house is appraised at less than Rs.20 lakhs. And it follows a
binomial distribution with n = 10, p=0.30 and q=0.70

27. A coin is tossed 2 times, if ‘X’ denotes the number of heads, find the probability distribution of
X.(Nov./Dec. 2013)

X: No. of heads 0 1 2

P(X=x) 1
2
1 1
2
1
2

   2C1   2C2  
2 4 2 2

28. If the probability that a target is destroyed on any one shot is 0.5, find the probability that it
would be destroyed on 6th attempt. (Nov./Dec.,2013)

P ( X  6)  q 5 p  (0.5)6

29. A continuous random variable X has the probability density function given by
f ( x)  a (1  x 2 ), 2  x  5 , Find ‘a’ and P( X  4) (May/June 2014)

5 4
1 1 31
2 a(1  x )dx  1  a  42 , P ( X  4)   (1  x 2 )dx 
2

2
42 63

30. For a binomial distribution with mean 6 and standard deviation 2 , Find the first two terms of
the distribution. (May/June 2014)

2 1
31. From the given data, n  9, p  , q
3 3
9 1 8
1  2 1
I term: 9C0   ; II term: 9C1    
 3  3  3

 x ;  1  x  1
32. Test whether f ( x)   probability density function of a continuous random variable
0 ; otherwise
(Nov./Dec.2014) (April/ May 2015)

1
1
 x2  1
1 x dx  2    2    1 . Therefore it is a continuous random variable.
 2 0 2

II Year / IV Sem 9
 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

33. What do you mean by MGF? Why it is called so? (Nov./Dec.2014)

MGF is a moment generating function which generates all the moments about the origin. It can also be
tr
calculated as a coefficient of and also by differentiating the MGF with respect to ‘t’ , r times , i.e.
r!
M X (t )  E (etx )
 dr 
r   r M X (t ) 
 dt  t 0

34. If the density function of a continuous random variable X is given by

ax ; 0  x  1
a ; 1  x  2

f ( x)   then find the value of ‘ a ’ (April/ May 2015)
3a  ax; 2  x  3
0 ; otherwise

1 2 3
1
 ax dx   a dx   (3a  ax)dx  1
0 1 2
a
2

35. Suppose that on an average, in every three pages of a book there is one typographical error. IF
the number of typographical errors on a single page of the book is a Poisson random variable.
What is the probability of at least one error on a specific page of the book? (April/ May 2015)

e
P ( X  1)  1  P( X  1)  1  P( X  0)  1   1  e 3
0!

36. What are the limitations of Poisson distribution? (April/ May 2015)

Poisson distribution is a limiting case of binomial distribution. When n   and p  0

Instead of Binomial distribution, Poisson distribution will be applied.

37. A continuous random variable X has the probability density function given by
f ( x)  a (1  x 2 ), 1  x  5 , Find ‘a’ and P( X  4) (Nov./Dec. 2015)

5
3
 a(1  x )dx  1  a 
2

1
136
4
3 18
P ( X  4)   (1  x 2 )dx 
1
136 34

II Year / IV Sem 10
 PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

38. What is meant by memory less property? Which discrete distribution follows this property?
(Nov./Dec. 2015)

If X is continuously distributed random variable, then for any two positive integers‘s’ and ‘t’
P ( X  s  t / X  s )  P ( X  t ) .Geometric distribution follows memory less property.

39. Let X be the random variable which denotes the number of heads in three tosses of a fair coin.
Determine the probability mass function of X? (Nov./ Dec. 2015)

X: No. of heads 0 1 2 3

P(X=x) 1 1
3
1
3
1
3
1
3

   3C1   3C2   3C3  

2 8 2 2 2

3
40. A continuous random variable X has a pdf given by f ( x)  (2 x  x 2 ), 0  x  2 .
4
Find P( X>1) (Nov./Dec. 2015)
2
3 1
P ( X  1)   (2 x  x 2 )dx 
1
4 2

x
41. Let X be a discrete random variable with pmf P( X  x)  , x  1, 2,3, 4 , Compute
10
X
P ( X  3) and E   (May/ June 2016)
2

Ans:
X: 1 2 3 4
1 2 3 4
p( x) :
10 10 10 10

1 2
P ( X  3) 
  0.3
10 10
X x  1  1   2   3  3   4  4 
E     p ( x )       1           
2 2  2   10   10   2   10   2   10 
3
42. If a random variable X has the moment generating function M X (t )  , Compute E ( X 2 )
3t
(May/ June 2016)
1 2
 t t t 1 t  2
2
3
M X (t )   1    1      .....  1  t       ...
3t  3 3 3  3  2!  9 
2
t
Coefficient of  E( X 2 )
2!
II Year / IV Sem 11
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

2
E( X 2 ) 
9
PART-B

1. A random variable X has the following probability distribution:

X=x -2 -1 0 1 2 3

P(x) 0.1 k 0.2 2k 0.3 3k

(i) Find k, (ii) Evaluate P X  2 and P 2  X  2 , (iii) Find the PDF of X and

(iv) Evaluate the mean of X (AP) (Nov/Dec 2011) (May/ June 2016)

2. A random variable X has the following probability distribution:

X=x -2 -1 0 1 2 3

P(x) 0.1 K 0.2 2K 0.3 3K

Find K, P 2  x  2 , mean of X . (AP) (May/Jun 2009)

3. A random variable X has the following probability function

X=x 0 1 2 3 4 5 6 7
2 2 2
P(x) 0 k 2 k 2 k 3 k k 2k 7k +k
(i) Find the value of , k (ii) Evaluate P  X  6  , P X  6  ,

(iii) If P X  c   , find the minimum value of c .

1
2
(iv) P (1.5  X  4.5 / X  2) (AP)(April/May 2012)(April/ May 2015)
4. If X is a Poisson variate such that P X  2  9 P X  4  90 P X  6 . Find
 
(i) Mean and E X 2 , (ii) P X  2. (AP)(April/May 2012)

5. In a continuous distribution, the probability density is given by f ( x )  kx ( 2  x ),0  x  2. Find

k, mean, variance and the distribution function. (AP) (May/Jun 2007)
6. The sales of a convenience store on a randomly selected day are X thousand dollars, where X is
a random variable with a distribution function of the following form:
0 x0
 2
 x 0  x 1
F ( x)   2
k ( 4 x  x 2 ) 1 x  2

1 otherwise
Suppose that this convenience store’s total sales on any given day are less than $2000.
(a) Find the value of k,
(b) Let A and B be the events that ‘tomorrow the store’s total sales are between 500 and
1500 dollars, and over 1000 dollars,’ respectively. Find P(A) and P(B).
II Year / IV Sem 12
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

(c) Are A and B independent events? (U) (Nov/Dec 2007)

2 x, 0  x 1  1
7. A random variable X has the pdf f ( x)   find (a) P X   (b)
0, otherwise  2
 3 
1 1 X  
P  X   (c) P 4  (AP) (Nov/Dec 2008)
4 2  1
X  
 2
 x
2


If p ( x )   xe , x  0 .Show that p(x) is a pdf and also find F(x) (AP) (May/Jun 2009)
2
8.
0, x  0
 4
1  2 , x2
9. If the cumulative distribution function of a RV X is given by F ( x)   x
0, x2
find (a) P X  3 (b) P4  X  5 (c) P X  3 (AP) (Apr/May 2008)
10. If the density function of a continuous random variable X is given
ax, 0  x 1
a 1 x  2

by f ( x)  
3a  ax 2 x3
o otherwise
Find a . Find the cdf of X . (AN) (Nov/Dec 2008)

11. The distribution function of a random variable X is given by F ( x)  1  (1  x)e  x ; x  0 .

Find the density function, mean and variance of X .(AP) (Nov/Dec 2010)
12. If X is a random variable with a continuous distribution function F (x ) , prove that
1
 ( x  1), 1 x  3
Y  F (x) has a uniform distribution in (0, 1). Further if f ( X )   2 ,
0 otherwise
find the range of Y corresponding to the range 1.1  x  2.9 .(AP)

13. The (DF) cumulative distribution function (cdf) of a random variable X is given by
0, x  0

 x 2 ,0  x  1
 2
F ( x)  
1 
 25
3
  1
3  x2 ,  x  3
2
1, x  3

 
Find the pdf of X and evaluate P X  1 and P 1  X  4 using both pdf and cdf (PDF).
3
(U)(May/June2007, (Nov/Dec 2011)
II Year / IV Sem 13
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

14. The probability function of an infinite discrete distribution is given by P X  j   1 j ;

2
j  1,2,...,  . Verify that the total probability is 1 and find the mean and variance of the
distribution. Find also P X is even, P X  5 and P X is divisible by 3. (AP) (Nov/Dec
2011)
15. Find the moment –generating function of the binominal random variable with parameters m
and p and hence find it mean and variance. (AN) (April/May 2011)(April/ May 2015)
16. Find the moment generating function of an exponential random variable and hence find its
mean and variance.(AN)(April/May 2012)(May/ June 2014)
17. Find the moment generating function of the geometric random variable with the pdf
f ( x)  pq x 1 , x  1,2,3,.... and hence obtain its mean and variance. (May/Jun 2007)(April/
May 2015) (AN)
18. Derive mean and variance of a Geometric distribution. Also establish the forgetfulness property
of the Geometric distribution. (AN)(April/May 2011)
19. Describe the situations in which geometric distribution could be used. Obtain its MGF. (AN)
(April/May 2010)
20. By calculating the MGF of Poisson distribution with parameter  , prove that the mean and
variance of the Poisson distribution are equal. (AN) (April/May 2010)(May/ June 2014)
21. Define Weibull distribution and write its mean and variance. (R)(April/May 2011)
22. Define Gamma distribution and find its mean and variance. (AN)(Nov/Dec 2011)
Ce 2 x 0  x  
23. If the density function of X equals f  x    . Find C . What is P X  2 . (AP)
0 x0
(April/May 2010)
5
1 3 
24. A discrete RV has moment generating function M X (t )    e t  . Find E  X  , Var  X 
4 4 
and P X  2 .(AP) (Apr/May 2008)
25.  
If the moments of a random variable X are defined by E X r  0.6; r  1,2,3,.. show that
P X  0  0.4, P X  1  0.6, P X  2  0 . (AN)(Nov/Dec 2008)
26. A coin having probability p of coming up heads is successively flipped until the r th head
appears. Argue that X , the number of flips required will be n , n  r with
r nr
probability P  X  n    n 1Cr 1 p q , n  r . (AN)(April/May 2010)
27. A coin is tossed until the first head occurs. Assuming that the tosses are independent and the
probability of a head occurring is ‘p’. Find the value of ‘p’ so that the probability that an odd
number of tosses required is equal to 0.6. Can you find a value of ‘p’ so that the probability is
0.5 that an odd number of tosses are required? (U)(Nov/Dec 2010)
28. The time (in hours) required to repair a machine is exponentially distributed with
1
parameter   . What is the probability that the repair time exceeds 2h ? What is the
2
conditional probability that a repair takes at least 10h given that its duration exceeds 9h . (AP)
II Year / IV Sem 14
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

(Nov/Dec 2010)
29. Suppose that telephone calls arriving at a particular switchboard follow a Poisson process with
an average of 5 calls coming per minute. What is the probability that up to a minute will elapse
unit 2 calls have come in to the switch board? (AP)(April/May 2011)
30. A machine manufacturing screws is known to produce 5% defective. In a random sample of 15
screws, what is the probability that there are
(i) exactly 3 defectives
(ii) not more than 3 defectives (AP)(Nov/Dec 2008)
31. Out of 800 families with 4 children each, how many families would be expected to have
(i) 2 boys and 2 girls
(ii) at least 1 boy
(iii) at most 2 girls
(iv) Children of both sexes.
Assume equal probabilities for boys and girls. (AP)(May/Jun 2009)

32. The mileage which cars owners get with a certain kind of certain kind of radial tire is a random
variable having an exponential distribution with mean 40,000 km. Find the probabilities that
one of these tires will last
(i) At least 20,000 km
(ii) At most 30,000 km. (AP)(April/May 2015)
33. The number of monthly breakdown of a computer is a random variable having a Poisson
distribution with mean equal to 1.8. Find the probability that this computer will function for a
month
(i) without a breakdown
(ii) with only one breakdown
(iii)With at least one break down (AP) (May/Jun 2007), (Nov. /Dec.2012)
34. Experience has shown that while walking in a certain park, the time X (in minutes) between
xe  x x  0
seeing two people smoking has a density function of the form f ( x )   .
0 otherwise
(a) Calculate the value of  .
(b) Find the distribution function of X .
(c) What is the probability that Jeff, who has just seen a person smoking, will see another
person smoking in 2 to 5 minutes? In atleast 7 minutes? (AP) (Nov/Dec 2007)
35. Let the random variable X follows binomial distribution with parameter n and p. Find
(1) probability mass function of X
(2) moment generating function
(3) mean and variance of X (AP)(Nov/Dec 2006)
36. The number of personal computers (PC) sold daily at a Computer world is uniformly
distributed with a minimum of 2000 PC and a maximum of 5000 PC. Find
(i) the probability that daily sales will fall between 2,500 and3,000 PC.
(ii) What is the probability that Computer world will sell at least 4000 PCs?
(iii)What is the probability that Computer world will sell exactly 2500 PCs?
II Year / IV Sem 15
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

(AP) (Nov/Dec 2006)

37. Define the probability density function of normal distribution and standard normal distribution.
Write the important properties of this distribution. (R)(Nov/Dec 2006)
38. An electrical firm manufactures light bulbs that have a life, before burn out, that is normally
distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find
(i) the probability that a bulb burns more than 834 hours.
(ii) the probability that the bulb burns between 778 and 834 hours. (AP) (Nov/Dec
2006)
39. The lifetime X in hours of a component is modeled by a Weibull distribution with   2 .
Starting with a large number of components, it is observed that 15% of the components that
have lasted 90 hrs fail before 100 hrs. Find the parameter  .(AP)(May/Jun 2007)
40. A die is cast until 6 appear. What is the probability that it must be cast more than 5 times?
(AP) (Nov/Dec 2008)
X
41. Let X be a RV with E  X   1 and E  X  X  1  4 . Find Var  ,Var (2  3 X ) (AP) (Apr/May
2
2008)
x 0  x 1
3

42. If X is a continuous RV with pdf f ( x)   ( x  1) 2 1  x  2 find the cumulative
2
0 otherwise
3 5
distribution function F x  of X and use it to find P  X   (AP)(Apr/May 2008).
2 2
43. If a RV X has geometric distribution, i.e., P ( X  x)  pq x 1 , x  1,2,3 where
q  1  p, o  p  1 show that P X  x  y X  y   P X  x  (AP)(Apr/May 2008)
 1  x2 2
 e ,x  0
44. Let the pdf for X be given by f ( x)   2 .
0 otherwise

 1
Find (i) P X   , (ii) Moment generating function for X (iii) E  X 
 2
(iv) Var( X ) (AP)(Apr/May 2008)
2(1  x) for 0  x  1
45. If the probability density of X is given by f ( x)  
0 otherwise

(i)  
Show that E X r 
2
(r  1)(r  2)
(ii) 
Use this result to evaluate E 2 x  1
2
 (AP)(Nov/Dec 2006)
1
 0 xk
46. A random variable X has density function given by f ( x)   k
0 otherwise
II Year / IV Sem 16
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

Find mgf, rth moment, mean, variance (AP)(Nov/Dec 2006)

47. Let the random variable X assume the value ‘r’ with the probability law:
P  X  r   q p, r  1,2,3,.. Find the moment generating function and hence its mean and
r 1

variance. (AN)(May/Jun 2006)

48. Find the moment generating function of a normal distribution. (AN) (May/Jun 2006)
49. The pdf of samples of the amplitude of speech wave forms is found to decay exponentially at
the rate  so the following pdf is proposed. f ( x)  Ce ,  x   . Find the constant C
 x

and also P  X    and E(X). (AP) (Apr/May 2008)

50. If X is uniformly distributed over (  ,  ),   0 , find  so that

P X  1  P X  1  P X  1 (AP) (Nov/Dec 008)

1
(i) (ii)
3
51. The lifetime of a TV tube (in years) is an exponential random variable with mean 10. What is
the probability that the average lifetime of a random sample of 36 TV tubes is at least 0.5
(AP) (Nov/Dec 2007)
52. The atoms of radioactive element are randomly disintegrating. If every gram of this element, on
average, emits 3.9 alpha particles per second, what is the probability that during the next
second the number of alpha particles emitted from 1 gram is
(i) at most 6 (ii) At least 2(iii) at least 3 and at most 6 (AP)(Nov/Dec 2007)
53. Starting at 5.00am every half hour there is a flight from San Francisco airport to Los Angeles
International Airport. Suppose that none of these planes is completely sold out and that they
always have room for passengers. A person who wants to fly to L.A. arrives at the airport at a
random time between 8.45 a.m., and 9.45., am. Find the probability that she waits
(i) At most 10 minutes (ii) At least 15 minutes (AP) (Nov/Dec 2007)
54. A man with ‘n’ keys wants to open his door and tries the keys independently and at random.
Find the mean and variance of the number of trials required to open the door if unsuccessful
keys are not eliminated from further selection. (AN) (Nov/Dec 2007)
55. Write the pdf of Gamma distribution. Find the MGF, mean and variance. (May/Jun 2007)
56. In a certain city, the daily consumption of electric power in millions of kilowatt hours can be
1
treated as a random variable having Gamma distribution with parameters   and v  3 . If
2
the power plant of this city has daily capacity of 12 million kilowatt-hours, what is the
probability that this power supply will be inadequate on any given day. (AP)(April/May 2012)
57. A continuous random variable has the pdf f ( x)  kx 4 ,  1  x  1 , Find the value of k and also
 1 1
P  X   X   (AP) (May/ June 2013)
 2  4 
58. Find the moment generating function of uniform distribution and hence find its mean and
variance. (AN) (May/ June 2013)
59. Find the moment generating function and rth moment for the distribution whose pdf is
f ( x)  Ke  x , 0  x  . Hence find the mean and variance. (AN) (May/ June 2013)
II Year / IV Sem 17
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

60. In a large consignment of electric bulb, 10 % are defective. A random sample of 20 is taken
for inspection. Find the probability that (1) all are good bulbs (2) atmost there are 3 defective
bulbs (3) Exactly there are 3 defective bulbs(May/ June 2013) (AP)
61. If a random variable X takes the values 1,2,3,4 such that P(X=1) = 3P(X=2) = P(X=3) =
5P(X=4). Find the probability distribution of X. (AP)(Nov. Dec.2012).
62. Find the MGF of the binomial distribution and hence find its mean? (AN)(Nov/ Dec. 2012)

63. If the probability that an applicant for a driver’s license will pass the road test on any trial is
0.8, what is the probability that he will finally pass the test (1) on the4th trial (2) in fewer than
4 trials? (AP) (Nov./ Dec.2012)
 x , for 0  x  1

64. Find the MGF of the random variable ‘X’ having the pdf f ( x)  2  x, for 1  x  2
0, otherwise

(AP) (Nov./Dec.2013)
65. A manufacturer of pins knows that 2% of his products are defective. If he sells pins in boxes of
100 and guarantees that not more than 4 pins will be defective, what is the probability that a
box fail to meet the guaranteed quality? (AP)(Nov./Dec.2013)
66. 6 dice are thrown 729 times. How many times do you expect atleast three dice to show a five
or a Six? (AP)(Nov./Dec.2013)
67. If a continuous RV X follows uniform distribution in the interval (0,2) and a continuous RV, Y
follows exponential distribution with parameter  such that P ( X  1)  P (Y  1) (AP)
(Nov./Dec.2013)
68. Suppose that a trainee soldier shoots a target in an independent fashion. IF the probability that
the target is shot on any one shot is 0.7. (1) What is the probability that the target would be
hit on tenth attempt? (2) What is the probability that it takes him less than 4 shots? (3) What
is the probability that it takes him an even number of shots? (AP)(May/ June2014)
69. Trains arrive at a station at 15 minutes intervals starting at 4 a.m. If the passenger arrive at a
station at a time that is uniformly distributed between 9.00 and 9.30, find the probability that
he has to wait for the train for (1) less than 6 minutes (2) more than 10 minutes (AP)(May/
June 2014)
 xe  x 2 , x  0
2

70. If f ( x)   then show that f ( x) is a pdf and find F(x) (AP)(Nov./Dec.2014)

0 ; x  0
71. Find the MGF of a poisson random variable and hence find its mean and variance (AN)
(Nov./Dec.2014)
1 1 1 1
72. A random variable X takes the values -2,-1, 0 and 1 with probabilities , , and
8 8 4 2
respectively. Find and draw the probability distribution function (AP)(Nov./Dec.2014)
73. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and
variance of the distribution. (AP)(Nov./Dec.2014)

II Year / IV Sem 18
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

74. The distribution function of a random variable X is given by F ( x)  1  (1  x)e  x , x  0 . Find

the density function, mean and variance of X. (AP) (April/May 2015)
75. Messages arrive at a switch board in a poisson manner at an average rate of six per hour. Find
the probability for each of the following events: (1) exactly two messages arrive within one
hour.(2) no message arrives within one hour (3) Atleast three messages arrive within one hour
(AP)(April/ May 2015)
76. The peak temperature T, as measured in degrees Fahrenheit, on a particular day is the Gaussian
(85,10) random variable. What is P(T>100), P(T<60) and P (70  T  100) (AP) (April/ May
2015)

0 , x  0

 1 1
77. The CDF of the random variable of x is given by F ( x)   x  , 0  x  Draw the graph of
 2 2
 1
1, x  2
1 1
CDF. Compute P ( X  1 ), P   X   (AP)(April/ May 2015)
4 3 2
78. Ten percent of the tools produced in a certain manufacturing company turn out to be defective.
Find the probability that in a sample of 10 tools chosen at random, exactly 2 will be defective
by using (1) binomial distribution (2) The Poisson approximation to the binomial distribution.
(AP) (Nov/Dec. 2015)
79. The number of typing mistakes that a typist makes on a given page has a Poisson distribution
with a mean of 3 mistakes. What is the probability that she makes (1) exactly 7 mistakes?
(2) fewer than 4 mistakes? (3) no mistakes on a given page? (AP) (Nov/Dec. 2015)
80. The lifetime X of particular brand of batteries is exponentially distributed with a mean of 4
weeks. Determine (1) the mean and variance of X (2) what is the probability that the battery life
exceeds 2 weeks? (3) Given that the battery has lasted 6 weeks, what is the probability that it
will last at least another 5 weeks? (AP) (Nov/Dec. 2015)
81. Find the moment generating function of N (  ,  2 ) normal random variable and hence determine
the mean and variance. (AN) (Nov/Dec. 2015)
82. A component has an exponential time to failure distribution with mean of 10,000 hours.
(1) The component has already been in operation for its mean life. What is the probability that
it will fail by 15,000 hours? (Nov/Dec. 2015)
(2) At 15,000 hours the component is still in operation. What is the probability that it will
operate for another 5000 hours? (AP) (Nov/Dec. 2015)
83. The average percentage of marks of candidates in an examination is 42 with a standard
deviation of 10. If the minimum mark for pass is 50% and 1000 candidates appear for the
examination, how many candidates can be expected to get the pass mark if the marks follow
normal distribution? If it is required, that the double the number of the candidates should pass,
what should be the minimum marks for pass? (AP) (Nov/Dec. 2015)

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

84. A continuous random variable X has the pdf f ( x)  kx 3e  x , x  0 . Find the rth moment about
the origin, moment generating function, mean and variance of X. (AP) (Nov/Dec. 2015)
85. Let X be a continuous random variable with pdf f ( x)  xe  x , x  0 , find (i) the cumulative
distribution function of X (ii) Moment generating function of X (iii) P(X<2) (iv) E(X) (AP)
(May/ June 2016)
x 1
 3  1 
86. Let P ( X  x)      , x  1, 2,3... be the probability mass function of a random variable
 4  4 
X, Compute (1) P(X>4) (2) P(X >4/ X>2) (3) E(x) (4) Var (X) (AP) (May/ June 2016)

87. Let X be a uniformly distributed random variable over [-5, 5], Determine (1) P(X<=2)
(2) P( X  2) , (3) Cumulative distribution function of X, (4) Var(X). (AP) (May/ June 2016)
Find the moment generating function of Poisson distribution with parameter  and hence prove that
the mean and variance of the Poisson distribution are equal. (AN) (May/ June 2016) (Nov/Dec.
2015)

COURSE OUTCOME: Acquire skills in handling one random variable and functions of random
variables.

UNIT – II
TWO DIMENSIONAL RANDOM VARIABLES

SYLLABUS: Joint distributions – Marginal and conditional distributions – Covariance – Correlation

and Linear regression – Transformation of random variables.

COURSE OBJECTIVE: Know the fundamental concepts of joint, marginal and conditional
distributions to understand covariance and correlation. Have an ability to design a model or a process
to meet desired needs within realistic constraints such as environmental conditions

PART–A
1. Define joint probability density function of two random variables X and Y .
If  X , Y  is a two dimensional continuous random variable such that
 dx dy 
  f  x , y  dx dy , then f x , y  is called the joint pdf
dx dy
Px   X  x  , y  Y  y 
 2 2 2 2 
of  X , Y  , provided f x , y  satisfies the following conditions
i  f x , y   0 for all x , y  R
ii   f x , y  dx dy  1
R

2. State the basic properties of joint distribution of  X , Y  where X and Y are random variables.

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

Statement:
Properties of joint distribution of  X , Y  are

i  F   , y  0  F x ,  and F   , 1
ii  Pa  X  b , Y  y  F b , y   F a , y 
iii  PX  x , c  Y  d  F x , d  F x , c
iv  Pa  X  b , c  Y  d  F b , d  F a , d  F b , c F a , c
2F
v  At po int s of continuity of f x , y ,  f x , y 
x y

3. Can the joint distributions of two random variables X and Y be got if their marginal
distributions are random?
Solution:
If the random variables X and Y are independent then the joint distributions of two random
variables can be got if their marginal distributions are known.

4. Let X and Y be two discrete random variable with joint pmf

x  2y
 , x  1, 2 ; y  1, 2
P X  x , Y  y   18 . Find the marginal pmf of X and EX .?
0 , otherwise

Solution:
The joint PMF of  X , Y  is given by 1 2

1 3 4
18 18

2 5 6
18 18
Marginal pmf of X is

PX  1 PX  2

3 5 8 4 4 6 10 5
   ,   
18 18 18 9 18 18 18 9

 4  5  4 10 14
EX   x px   1   2     .
9 9 9 9 9

5. Let X and Y be integer valued random variables with

P X  m , Y  n   q p
2 mn2
, n , m  1, 2 , ............ and p  q  1 . Are X and Y independent?

Solution:

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

The marginal PMF of X is

  
px    q 2 p m  n  2   q 2 p m 1 p n 1  q 2 p m 1  p n 1
n 1 n 1 n 1

q p 2 m 1
1  p  p 2

 p  .........  q p
3 2 m 1
1  p  1
 q 2 p m 1 q 1  q p m 1

The marginal PMF of Y is

  
p y    q 2 p m  n  2  q 2
p m 1 p n 1  q 2 p n 1 p m 1

m 1 m 1 m 1


 q 2 p n  1 1  p  p 2  p 3  .........  q 2 p n  1 1  p   1 
 q 2 p n 1 q 1  q p n 1

p  x  p  y   q p m  1 . q p n  1  q 2 p m  n  2  P X  m Y  n 

Therefore X and Y are independent random variables.

6. The joint probability density function of the random variable X ,Y  is given by

f  x , y   k x y e  x y  , x  0 , y  0 . Find the value of k.
2 2
(Nov./Dec.2013)
Solution:
Given f x , y  is the joint pdf , we have

 f x , y  dx dy 1 put x 2  t

 x  dx dy 1
k x ye
2
 y2
2 x dx  dt
0 0

dt
 x ye x dx 
 x2
e  y dx dy  1
2
k
0 0
2

 
k  y e   x e  x dx  dy  1
 y2
when x  0 , t  0 and when x   , t  
2

0 0 
2  dt 
 
k  y e  y   e t  dy  1
0 0 2


 y e  e 
k  y2 t 
0 dy  1
2 0

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

put y 2  t
2 y dy  dt
dt
y dy 
 2
 y e 0  1dy
k  y2
1 when y  0 , t  0 and when y   , t  
2 0

k   t dt
20
e
2
k
 1  et
4
  
0 1 
k
4
0  1  1  k  1  k  4 .
4

k x  y  , 0  x  2 ; 0  y  2
7. The joint PDF of the random variable  X , Y  is f x , y    .
0 , otherwise

Find the value of k .

Solution:
Given f x , y  is the joint pdf , we have

2 2 2  2 
2 
 f x , y  dx dy  1    k x  y  dx dy  1  k   x
 2
  y  x 02  dy  1

0 
0 0 0  
2 2
k  2  0  y 2  0 dy  1  k  2  2 y  dy  1
0 0

  y2  
2

 k  2  y 02  2     1 k  1
  2   8
  0 

c x y ,0 x  2 ;0 y  2
8. The joint pdf of the random variable  X , Y  is f x , y    . Find the
0 , otherwise
value of c .

Solution:
Given f x , y  is the joint pdf , we have

2
 x2 
2 2 2 2

 f  x , y  dx dy  1   
c x y dx dy  1  c y   dy  1  c y 2  0  dy  1

 2 
0 0 0  0 0

2
 y2  1
 2c    1  4c  1  c 
  0
2 4

9. If two random variables X and Y have probability density function f x , y   k 2 x  y  for

0  x  2 and 0  y  3 . Evaluate k .

Solution: Given f x , y  is the joint PDF, we have

II Year / IV Sem 23
PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

3 2 3   2 2 
 f  x , y  dx dy  1 
 k 2 x  y  dx dy  1  k  2 
  y  x 02  dy  1
x 
  2  
0 0 0  0 
3   y2 
3

 k 4  2 y  dy  1  k  4  y 30  2 
   1
  2  
0   0 

 k 12  9  1  21 k  1  k 
1
21

10. If the function f x , y   c 1  x 1  y , 0  x 1 , 0  y 1 is to be a density function, find the value

of c.

Solution:

Given f x , y  is the joint PDF, we have

1 1 1 1

 f  x , y  dx dy  1    c 1  x 1  y  dx dy  1  c   1  x  y  xy  dx dy  1
0 0 0 0

 1  x2
1

1
 x2  
1

 c   x 0     y  x 10  y   dy  1
0 
  2 0  2  0 
 1 y 1 y
1 1
 c  1   y   dy  1  c     dy  1
0 
2 2 0
2 2
 1 1 1  y 2 1  1 1
 c   y 0      1  c     1   1  c  4
c
 2 2  2 0 
 2 4 4
Therefore the value of c is c  4

Find the marginal density functions of X and Y if f x , y   2 x  5 y , 0  x 1, 0  y 1 .

2
11.
5
Solution:

Marginal density of X is

2  y2  
1 1

f X  x    f  x , y  dy   2 x  5 y  dy   2 x  y 0  5   
2 1

50 5  2  0 

2 5 4
 2 x    x  1, 0  x  1
5 2 5

Marginal density of Y is

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

2   x2  
1 1

f Y  y    f  x , y  dx   2 x  5 y  dx   2    5 y  x 0 
2 1

50 5   2 0 

 1  5 y    2 y , 0  y  1
2 2
5 5

 x  y ; 0  x 1 , 0  y 1
12. If X and Y have joint pdf f x , y    . Check whether X and Y are
0 ; otherwise
independent.

Solution:
1
1
 y2 
f X  x    f  x , y  dy    x  y  dy  x  y 0  
1
1
  x  , 0  x  1
0  2 0 2

1
1
 x2 
f Y  y    f  x , y  dx    x  y  dx     y  x 10  y  , 0  y  1
1
0  2 0 2

 1  1
f X x . f Y  y    x    y    xy     x  y  f x , y 
x y 1
 2  2 2 2 4

Therefore X and Y are not independent variables.

13. If X and Y are random variables having the joint density function

6  x  y , 0  x  2 , 2  y  4 , find PX  Y  3 .
f x , y  
1
8
Solution:

P  X  Y  3 
 f  x , y  dx dy

33  y 3 
 2 3  y 
 3 y
   
1 1 x 
 6  x  y dx dy   6  y   x 0 
 2 
 dy
8 8  
2 0 2   0 

3 3
  1  
 
1 1 1
  6  y  3  y  3  y 2  dy  18  9 y  y 2  3  y 2  dy
8  2  8  2 
2 2
 
 2 3  3 3  3
1  1   3  y 3  
9 
y   y 
 18  y 3    
8  2  2   3  2   3  
  2  2 
 
2 
 
1 9 1 1 
 18  3  2   9  4   27  8    0  1 
8 2 3 6 
1  45 19 1 5
 18     
8  2 3 6 24

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PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

14. Let X and Y be continuous random variable with joint pdf

f XY x , y   
3 2
2

x  y 2 , 0  x 1 , 0  y 1 . Find f X Y x y  .

Solution:
3  x 3  3 1
1

   3
1
f Y  y    f  x , y  dx   x 2  y 2 dx     y 2  x 10     y 2   y 2 
3 1
20 2  3
 0  2  3  2 2

f x , y  2
3 2

x  y2
x2  y2

f X Y x y     .
fY  y 3  2 1 1
y   y 
2

2 3 3

15. If the joint pdf of  X , Y  is given by f x , y   2  x  y ; 0  x  y 1 , find EX .

Solution:

E  X    x f  x , y  dx dy    x 2  x  y  dx dy    2 x  x 2  xy dx dy
1 y 1 y

0 0 0 0

  x2
1 y
  x3 
y
 x2  
y

   2       y    dy
0 
  0  3 0  2  0 
2
1 1
1
 2 y3 y3  1
 5   y3  5  y4 
   y    dy    y 2  y 3  dy      
0
0
3 2  6   3 0 6  4 0
1 5 3 1
   
3 24 24 8

16. Let X and Y be random variable with joint density function

4 x y ; 0  x 1 , 0  y 1
f XY x , y    . Find E  X Y  .
0 , otherwise

Solution:
1
1 1 1 1
 x3 1

E  XY    xy f  x , y  dx dy    xy 4 x y  dx dy  4   x y dx dy  4  y 
2 2 2
 dy
0 0 0 0 0  3 0
1
4
1
4 y3  4  1  4
 0 y dy  3      .
2

3  3 0 3  3  9
17. Let X and Y be any two random variables and a ,b be constants. Prove that
Cov a X , bY   ab cov  X , Y 
Solution:
Cov  X , Y   E XY  EX  EY 

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Cov a X , b Y   E aX bY   EaX  EbY 

 ab EX Y   a EX  b EY   ab E  XY   E  X  E Y 
 a b Cov  X , Y 

18. If Y   2 X  3 , find Cov  X , Y  .

Solution:
Cov  X , Y   E XY  EX  EY 
 E X  2 X  3  EX  E 2 X  3
 
 E  2 X 2  3 X  EX  2 EX  3
 
  2 E X 2  3 EX  2 EX   3 EX 
2

  2 E X  EX     2Var X
2 2

19. If X 1 has mean 4 and variance 9 while X 2 has mean  2 and variance 5 and the two are
independent, find Var 2 X 1  X 2  5 .
Solution:
Given EX 1  4 ,Var X 1  9

EX 2   2 ,Var X 2  5

Var 2 X 1  X 2  5  4Var X 1  Var X 2  4 9  5  36  5  41

20. Find the acute angle between the two lines of regression.
Solution:

The equations of the regression lines are

y
y y  r x  x             1
x
x
xx  r  y  y             2
y

y
Slope of line 1 is m1  r
x

y
Slope of line 2  is m2 
r x

If  is the acute angle between the two lines, then

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m1  m 2
tan  
1  m1 m 2
y

y r 2

1  y

1  r   y2

1  r  x  y
r
 x r x r x r x 2
   
y y  y2  x2  y 2 r  x   y 
2 2
1 r . 1
 x r x  x2  x2

21. If X and Y are random variables such that Y  a X  b where a and b are real constants, show
that the correlation co-efficient r  X , Y  between them has magnitude one.
Solution:
Cov  X , Y 
Correlation co-efficient r  X , Y  
 X Y

Cov  X , Y   E XY  EX  EY   E X a X  b   EX  Ea X  b

 
 E a X 2  b X  EX a EX  b 
 
 a E X 2  b EX  a EX   b EX 
2

 a E X  EX    a Var X  a 
2 2 2
X

2
 
 Y  E Y 2   EY  
2

  
 E aX  b    EaX  b   E a 2 X 2  2ab X  b 2  a EX  b 
2 2
 2

 
 a 2 E X 2  2ab EX  b 2  a 2 EX   2ab EX  b 2
2

   
 a 2 E X 2  EX   a 2 Var X  a 2  X
2 2

a X
2

Therefore  Y  a X and r  X , Y   1
 X .a X

Therefore, the correlation co-efficient r  X , Y  between them has magnitude one.

22. If X and Y are two independent random variables with variances 2 and 3, find the variance of
3X+4Y (May/ June 2013)

Solution: Var (3 X  4Y )  9Var ( X )  16 Var (Y )  9(2)  16(3)  66

23. If the joint pdf of (X,Y) is given by f ( x, y )  2, 0  x  y  1 , Find E(X) (May/ June 2013)

Solution:
1
The marginal density function of X is f ( x)   f ( x, y ) dy  2(1  x)
x

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1 1
1
E ( X )   x( f ( x)dx   x(2)(1  x)dx 
0 0
3

24. When will the two regression lines be (A) at right angles (b) coincident? (Nov./ Dec. 2012)
Solution:
If r  1 , the regression lines will coincide.
If r  0 , the regression line will be at right angle to each other.

25. A small college has 90 male and 30 female professors. An ad-hoc committee of 5 is selected at
random to unite the vision and mission of the college. If X and Y are the number of men and
women in the committee, respectively. What is the joint probability mass function of X and Y?
(Nov./ Dec. 2012)

X=No. of
men
Probability
Y= No. of
female
X=5, Y=0 9
5

 5C 5  
 12 
X=4, Y=1 9 3
4 1

 5C4    
 12   12 
3 2
X=3, Y=2 9 3
 5C3    
 12   12 
X=2, Y=3 9 3
2 3

 5C 2    
 12   12 
X=1, Y=4  9  3
1 4

 5C1    
 12   12 
X=0, Y=5 9 3
0 5

 5C 0    
 12   12 

26. Find the value of ‘k’ if the joint density function of (X, Y) is given by
f ( x, y )  k (1  x)(1  y ), 0  x  4, 1  y  5 (May/ Nov. 2014)
4 5
1
  k (1  x)(1  y)dxdy
0 1
 1 k 
32
1
27. Given the joint probability density function of (X,Y) as f ( x, y )  , 0  x  2, 0  y  3
6
Determine the marginal density. (May/June 2014)

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2 3
1 1 1 1
f ( y )   dx  ; f ( x)   dy 
0
6 3 0
6 2
28. The joint pdf of a two dimensional random variable (X,Y) is given by
kxe , 0  x  2, y  0
y
f ( x, y )   Find the value of ‘k. (April / May 2015)(Nov./Dec.2014)
 0, otherwise
2
2    y  1
0 0 kxe dy dx  k  0 x dx  
y
e dy   1  k 
 0  2
29. In a partially destroyed laboratory record of an analysis of correlation data, the following
results only are legible: Variance of x = 9; Regression equations are
8 X  10Y  66  0 and 40 X  18Y  214  0 . What are the mean values of X and Y?
(April/ May 2015)
8 X  10Y  66  0
Solving the equations
40 X  18Y  214  0
Mean value of X is 13 and the mean value of Y is 17.
30. The joint probability mass function of a two dimensional random variable (X,Y) is given by
p ( x, y )  k (2 x  3 y ); x  0,1, 2; y  1, 2,3. Find the value of k. (April/ May 2015)
X 0 1 2
Y
1 3k 5k 7k
2 6k 8k 10k
3 9k 11k 13k
1
k
72
31. What do you mean by correlation between two random vairble ? (April/ May 2015)
When the random variables X and Y are correlated, the correlation coefficient between X and
Y lies between -1 and +1
If the correlation coefficient is 0, then the random variable are uncorrelated.
If the correlation coefficient is -1, the random variables are negatively correlated.
If the correlation coefficient is +1, the random variables are positively correlated.
32. Given the two regression lines 3 X  12Y  19 and 3Y  9 X  46 , Find the coefficient of
correlation between X and Y ? (Nov./Dec. 2015)
1 1
From the first equations byx     and from the second equation bxy    
4 3
1
Then the correlation coefficient between x and y is  xy   bxy byx  
 0.08
12
33. The joint probability density function of bivariate random variable (X,Y) is given by
f ( x, y )  4 xy, 0  x  1, 0  y  1 , Find P ( X  Y  1) (Nov./Dec. 2015)

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1 1 x
1
P ( X  Y  1)    4 xy dy dx  6
0 0

34. Determine the value of the constant ‘c’ if the joint density function of two discrete random
variables X and Y is given by p (m, n)  cmn, m  1, 2,3 and n  1, 2,3 (Nov./Dec. 2015)
49
35. The lines of regression in a bivariate distribution are X  9Y  7, and Y  4 X  , Find the
3
correlation coefficient? (Nov./Dec. 2015)

36. Comment on the statement: “If COV(X,Y)=0,then X and Y are uncorrelated”. (Nov./ Dec.2014)
Since X and Y are uncorrelated, the correlation coefficient between them is zero. Therefore x and Y are
independent random variables.
1
37. The joint pdf of RV (X,Y) is given as f ( x, y )  , 0  y  x  1 , Find the marginal pdf of Y.
x
(May/ June 2016)
1
1 1
f y ( y )   dx  log  
y
x  y
38. Let X and Y be two independent random variables with Var (X)=9, Var(Y)=3, Find the
Var(4X-2Y+6) (May/ June 2016)
Var (4 X  2Y  6)  16 Var ( X )  4 Var (Y )  16 cov( X , Y )
 16(9)  4(3)  0  144  12  156

PART-B

1. Let X and Y have the joint pdf

X

0 1 2
Y
0 0.1 0.4 0.1

1 0.2 0.2 0

Find

(i) P X  Y  1
(ii) the probability mass function P X  x  of the RV X
(iii) PY  1 X  1
(iv) E  XY  (AP)(Apr/May 2008)
2. The joint density function of the random variable  X , Y  is given by

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8 xy,0  x  1,0  y  x
f ( x, y)  
0, elsewhere

(i) Find the marginal density of Y

(ii) Conditional density of X  y
Y
 1
(iii) P X   (AP)(May/Jun 2007)
 2
3. The joint probability density function of a two dimensional random variable  X , Y  is given by
x2
f ( x, y )  xy 2  ;0  x  2,0  y  1
8
 
Compute P X  1, P Y  1 , P X  1  
, P Y 
1 
X  1, P X  Y , P( X  Y )  1 .
 2  1   2 
Y 
 2
(AP) (May/Jun 2009) (Nov/Dec 2007)
4. Given f ( x, y )  cx ( x  y ) , 0  x  2 ,  x  y  x and ‘0’ elsewhere. Evaluate ‘c’ find
f X x  and f Y  y  (AP)(Nov/Dec 2010)
5. In producing gallium – arsenide microchips, it is known that the ratio between gallium and
arsenide is independent of producing a high percentage of workable water, which are main
components of microchips. Let X denote the ratio of gallium to arsenide and Y denote the
percentage of workable micro wafers retrieved during a 1 hour period. X and Y are independent
random variables with the joint density being known as
 x(1  3 y 2 )
 0  x  2,0  y  1
f ( x, y )   4
0
 otherwise

Show that E  XY   E  X E Y  . (AP)(Nov/Dec 2006)

 x(1  3 y 2 )
6. Given the joint density function f ( x, y )   0  x  2, 0  y  1 , Find the marginal
4
0
 elsewhere
densities g ( x ) , h( y ) and the conditional density f ( x / y) and evaluate
1 1 
P   x  / Y  1 / 3 (AP)(Apr/May2011)
4 2 
7. The joint pmf of  X , Y  is given by p ( x , y )  k ( 2 x  3 y ), x  0,1, 2; y  1, 2,3 . Find all the
marginal and conditional probability distributions. Also find the probability distribution
of  X  Y  . (AP)(Nov/Dec 2007) (Nov/Dec 2011)

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8. Let X and Y be two random variables having the joint probability function
f ( x , y )  k ( 2 x  3 y ). where x and y can assume only the integer values 0, 1 and 2. Find all
the marginal and conditional distributions. (AP)(April/May 2012)
6e 3 x1  2 x2
for x1  0, x 2  0
9. If the joint density of X 1 , X 2 is given by f ( x1 , x 2 )   find the
0 otherwise
probability density of Y  X 1  X 2 . (AP)(Nov/Dec 2006)
10. Let X and Y be independent random variables, both uniformly distributed on (0, 1). Calculate
the probability density of X  Y . (AP)(April/May 2010)
11. Two random variables X and Y have the joint density
xy
function f XY ( x, y )  x 2  0  x  1,0  y  2 . Find the conditional density functions.
3
Check whether the conditional density functions are valid. (AP)(Nov/Dec 2006)
12. Suppose that X and Y are independent non-negative, continuous random variables having
densities f X x  and f Y  y  respectively. Compute PX  Y  (AP)(April/May 2010)
 1  xy
 ye 0  x   ,0  y  2
13. The joint density of X and Y is given by f ( x, y )   2 .
0 otherwise
Calculate the conditional density of X given Y  1 . (AP)(April/May 2010)
14. Determine whether the random variables X and Y are independent , given their joint
 2 xy
x  , 0  x  1, 0  y  2
probability density function as f ( x, y )   3
0 otherwise
(April/May 2011)
15. Can Y  5  2.8 X , X  3  0.5Y be the estimated regression equations of Y on X and X on
Y respectively? Explain your answer with suitable theoretical arguments.
(AP)(Nov/Dec 2007)
16. Two random variables X and Y have the following joint probability density function
2  x  y 0  x  1,0  y  1
f ( x, y )  
0 otherwise
Find
(a) Marginal probability density functions of X and Y
(b) Conditional density functions
(c) Var  X  and Var Y  . (AP)(May/June 2006)
17. Two random variables X and Y have the following joint probability density function
c(4  x  y ) 0  x  2, 0  y  2
f ( x, y )   Find cov X , Y  . Find the equations of two
0 elsewhere
lines of regression.(AP)(Apr/May 2012) (Nov/Dec. 2015)

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18. If X and Y are independent random variables with pdf’s e  x , x  0 and e  y , y0
X
respectively, find the density functions of U  and V  X  Y . Are U and
X Y
V independent?(AP) (Nov/Dec 2011)
19. Let  X , Y  be a two dimensional non negative continuous random variable having the joint
4 xye  ( x  y ) , x  0, y  0
2 2

density f ( x, y )   Find the density function of U  X 2  Y 2

0 elsewhere
(AP) (May/Jun 2006)
20. Two dimensional random variable  X , Y  have the joint probability density function
8 xy, 0  x  y 1  1 1
f ( x, y )   (i) Find P  X   Y   , (ii) Find the marginal and
0 elsewhere  2 4
conditional distributions. (iii) Are X and Y independent. (AP)(April/May 2012)
21. If the joint probability density of X1, X 2 is given by
e  ( x1  x2 ) for x1  0, x 2  0 X1
f ( x1 , x 2 )   . Find the probability of Y 
0 elsewhere X1  X 2
(AP)(Nov/Dec 2006)
22. If X and Y are independent random variables having density functions
 2 e 2 x , x0 and 3e 3 y , y0
f ( x)   f ( y)   respectively, find the density
0 x0 0 y0
functions z  X  Y . (AP)(Apr/May 2011)
23. If the joint distribution function of X and Y is given by
(1  e  x )(1  e  y ) for x  0, y  0
F ( x, y )  
0 otherwise

(i) Find the marginal densities of X and Y

(ii) Are X and Y independent
(iii) Find P (1  x  3,1  Y  2 ) (AP)(Nov/Dec 2008)
 1  x2 2
24. Let X be a random variable with pdf f ( x)   e ,  x   , find the pdf of the RV
 2
Y  X2. (AP)(Apr/May 2008)
25. Compute the co-efficient of correlation between X and Y using the following data
(AP)(Nov/Dec 2010)

X 1 3 5 7 8 10

Y 8 12 15 17 18 20

26. If the correlation coefficient is 0, then can we conclude that they are independent? Justify your
answer through an example. What about the converse? (AN)(April/May 2010)
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27. Find the coefficient of correlation between industrial production and export using the following
data: (AP)(Nov/Dec 2008)
Production( X ) 55 56 58 59 60 60 62

Export( Y ) 35 38 37 39 44 43 44

28. Obtain the equations of the regression lines from the following data, using the method of least
squares. Hence find the coefficient of correlation between X and Y . Also estimate the value
of Y when X  38 and the value of X when Y  18 . (AP)(May/Jun2009)(April/May 2015)

X 22 26 29 30 31 33 34 35

Y 20 20 21 29 27 24 27 31

29. Calculate the correlation coefficient for the following data: (AP) (May/Jun 2007) (Nov/Dec
2007)
X 65 66 67 67 68 69 70 72

Y 67 68 65 68 72 72 69 71

30. Find the coefficient of correlation and obtain the lines of regression from the data given below:
(AP)(Nov/Dec 2006)
X 50 55 50 60 65 65 65 60 60 50

Y 11 14 13 16 16 15 15 14 13 13

31. Find the coefficient of correlation and obtain the lines of regression from the data given below:
(AP)(May/Jun 2006)
X 62 64 65 69 70 71 72 74

Y 126 125 139 145 165 152 180 208

1 1
32. Let the random variable X have the marginal density f1 ( x)  1, x and let the
2 2
 1
1 x  y  x  1,  x0
conditional density of Y be f  y x    2 . Show that the variables are

1 1
x  y  1  x 0  x 
 2
uncorrelated. (AP)(May/Jun 2006)
1
33. Let z be a random variable with probability density f ( z ) 
in the range  1  z  1 . Let the
2
random variable X  z and the random variable Y  z 2 . Obviously X and Y are not
independent since X 2  Y . Show, none the less, that X and Y are uncorrelated. (AP)
(Nov/Dec 2006)
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34. Two random variables X and Y are defined as Y  4 X  9 . Find the correlation coefficient
between X and Y . (AP)(Nov/Dec 2006)
35. If X and Y are independent exponential random variables each with parameter 1, find the pdf of
U  X Y . (AP)(May/Jun 2007)(May/ June 2013) (Nov/Dec 2015)
2 x,0  x  1
36. If X is any continuous RV having the pdf f ( x)   and Y  e  X , find the pdf of
0, otherwise
RV Y . (AP)(Apr/May 2008)
e  ( x  y ) , x  0, y  0
37. Find P X  2 Y  4 when the joint pdf of X and Y is given by g ( x, y )   .
o otherwise
X
Are X and Y independent RVs? Explain. And find the pdf of the RV U  (AP)
Y
(Apr/May 2008)(May/ June 2016)
1, 0  x  y 1
38. If the joint pdf of the RVs X and Y is given by f ( x, y )   find the pdf of
0 otherwise
X
the RV U  (AP)(Apr/May 2008)
Y
8 x  10 y  66  0
39. The two lines of regression are The variance of X is 9.
40 x  18 y  214  0 .
Find the mean values of X and Y , Correlation coefficient between X and Y . (AP)(Nov/Dec
2008)
40. For two random variables X and Y with the same mean, the two regression equations are
y  ax  b and x  cy  d . Find the common mean, ratio of the standard deviations and also
b 1 a
show that  . (AP)(Nov/Dec2010)
d 1 c
41. The joint probability density function of a two dimensional random variable (X,Y) is
6 x y
f ( x, y )  , 0  x  2, 2  y  4 .
8
Find (1) P ( X  1  Y  3) ( 2) P ( X  Y  3) (3) P ( X  1 / Y  3) (AP) (May/ June 2013)
42. The marks obtained by 10 students in Mathematics and Statistics are given below. Find the
correlation coefficient between the two subjects (AP) (May/ June 2013)

Marks in Mathematics 75 30 60 80 53 35 15 40 38 48

Marks in Statistics 85 45 54 91 58 63 35 43 45 44

43. A distribution with unknown mean  has variance equal to 1.5, Use central limit theorem to
find how large a sample should be taken from the distribution in order that the probability will
be atleast 0.95 that the sample mean will be within 0.5 of the population mean. (AP)
(May/ June 2013)
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44. Obtain the equation of the lines of regression from the following data: (AP) (Nov./ Dec. 2012)
X: 1 2 3 4 5 6 7
Y: 9 8 10 12 11 13 14
xy 2 , 0  x  y  1
45. The joint pdf of random variable X and Y is given by f ( x, y )  
0 ; otherwise
(1) Determine the value of 
(2) Find the marginal probability density function of X and Y
(3) Find the conditional pdf f(x/y) (AP) (May/ June 2016)(Nov./ Dec. 2012)
46. The regression equations of X and Y is 3 y  5 x  108  0 . If the mean value of Y is 44 and the
9
variance of X were th of the variance of Y. Find the mean value of X and the correlation
16
coefficient. (AP) (Nov. / Dec. 2012)
47. Let X 1 , X 2 ,...., X 100 be independent identically distributed random variables with   2
1
And  2  . Find P (192  X 1  X 2  ...  X 100  210) . (AP) (Nov./Dec.2012)
4
54. Let X and Y be random variables having joint density function
3 2
 ( x  y ), 0  x  1, 0  y  1
2
f ( x, y )   2
0, elsewhere
Find the correlation coefficient  xy (AP)(Nov./Dec.2013)
x y
48. The joint distribution of X and Y is given by f ( x, y )  , x  1, 2,3, y  1, 2 . Find the
21
marginal distributions and conditional distributions. (AP) (Nov./Dec.2013) (Nov/Dec. 2015)
49. If the pdf of ‘X’ is f ( x)  2 x,  0  x  1 , find the pdf of Y= 3X+1 (AP) (Nov./Dec.2013)
50. The joint probability density function of two random variables X and Y is given b
6 xy 
f ( x, y )   x 2   , 0  x  1, 0  y  2 , Find the conditional density function of X
7 2 
given Y and the conditional density function of Y given X (AP)(May/ June 2014)
51. If the independent random variables X and Y have the variances 36 and 16 respectively. Find
the correlation coefficient ruv , where U =X+Y and V= X-Y (AP)(May/ June, 2014)
52. The joint probability density function of two random variables X and Y is
f ( x, y )  k ( x  y )  ( x 2  y 2 )  , 0  ( x, y )  1 , Show that X and Y are uncorrelated but
not independent? (AP)(May/ June 2014)
53. Calculate the coefficient of correlation for the following data: (AP)
X: 9 8 7 6 5 4 3 2 1
Y: 15 16 14 13 11 12 10 8 9 (Nov./Dec. 2014)

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54. IF X 1 , X 2 ,...., X n are Poisson variates with parameter   2 , Use the central limit theorem to
estimate P (120  S n  160) where S n  X 1  X 2  ...  X n , n  75 (AP)(Nov./Dec.2014)
55. The joint probability density function of a two dimension random variable (X,Y) is given by
x2
f ( x, y )  xy 2 ; 0  x  2; 0  y  1 .Compute
8
1  1  1 
P ( X  1), P (Y  ), P  X  1/ Y   , P  Y  / X  1 , P ( X  Y )and P ( X  Y  1)
2  2  2 
(AP)(April/ May 2015)
56. Find the equation of the regression line Y on X from the following data:
X: 3 5 6 8 9 11
Y: 2 3 4 6 5 8 (AP)(April/ May 2015)
57. Assume that the random variable X and Y have the joint
1
PDF f ( x, y )  x 3 y; 0  x  2; 0  y  1 Determine if X and Y are independent (April/ May
2
2015) (AP)
58. The joint pdf of the random variable X and Y is defined as f ( x, y )  25e 5 y ; 0  x  0.2, y  0
(1) find the marginal PDFs of X and Y (2) what is the covariance of X and Y? (April/ May
2015) (AP)
59. Find the constants k such that f ( x, y )  k (1  x)e  y , 0  x  1, y  0 is the joint pdf of the
continuous random variable (X, Y), Are X and Y independent r.v’s Explain. (May/ June 2016)
(AP)
x y
60. The joint distribution of X and Y is given by f ( x, y )  , x  1, 2, y  1, 2,3, 4 . Compute
32
the covariance of X and Y . (AP)(May/ June 2016)
61. Let the joint pdf of (X,Y) be given as f ( x, y )  Cxy , 0  x  y  1 , Determine the value of C ,
2

Find the marginal pdf of X and Y and find the conditional pdf f(x/ y) (AP)(May/ June 2016)
62. If X and Y are independent random variables with pdf’s e  x , x  0 and e  y , y  0 respectively,
X
find the density functions of U  . (AP) (Nov/Dec 2011) (Nov/Dec 2015)
X Y
63. The probability density function of X and Y is given by
6 xy 
f ( x, y )   x 2  , 0  x  1, 0  y  2 (1) compute the marginal density function of X and Y,
7 2 
 1 1
(2) Find E(X) and E(Y) (3) Find P  X  , Y   , 0  x  1, 0  y  2 . (AP)(Nov/Dec 2015)
 2 2
64. If X, Y and Z are uncorrelated random variables with zero means and standard deviation 5, 12
and 9 respectively and if U = X + Y and V = Y + Z, find the correlation coefficient between U
and V. (AP)(Nov/Dec 2015)
65. If X1, X2,…… Xn are Poisson variates with parameter   2 , use CLT to estimate P(120 < Sn
< 160) where Sn = X1 + X2 +,……+ Xn and n = 75. (AP)Nov/Dec 2015)
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COURSE OUTCOME: Acquire skills in handling more than one random variable and correlation
between the random variables.

UNIT III

CLASSIFICATION OF RANDOM PROCESSES

SYLLABUS: Classification – Stationary process – Markov process - Poisson process – Discrete

parameter Markov chain – Chapman Kolmogorov equations – Limiting distributions.

COURSE OBJECTIVE: Ability to analyze the relation between random input and output signals
using the basics of random process and its characteristics and to solve problems and model situations
using techniques of Markov process. Have an ability to design a model or a process to meet desired
needs within realistic constraints such as environmental conditions

PART A

1. State the four types of stochastic processes.

The four types of stochastic processes are
1. Discrete random sequence
2. Continuous random sequence
3. Discrete random process
4. Continuous random process
2. Give an example for a continuous time random process.
Solution:
If X (t ) represents the maximum temperature at a place in the interval (0, t), X (t ) is a
continuous random process.

3. Define a stationary process.

Solution:
If certain probability distribution or averages do not depend on t, then the random process
X (t ) is called a stationary process.
4. Give an example for a stationary process.
A Bernoulli process is a stationary process.
5. Give an example of stationary process and justify your claim.
Solution:
A Bernoulli process is a stationary stochastic process as the joint probability distributions are
independent of time.

6. Define strict sense and wide sense stationary process. (May/ June 2013)(May/June ,2014)
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Solution:
A random process is called a strict sense stationary process or strongly stationary process if all
its finite dimensional distributions are invariant under translation of time parameter.

A random process X (t ) with finite first and second order moments is called a weakly
stationary process or covariance stationary process or wide-sense stationary process if its mean
is a constant and the auto correlation depends only on the time difference. i.e, if E[ X (t )]  
and E[ X (t ) X (t  z )]  R ( z )

7. Give an example for strict sense stationary process.

Bernoulli’s process is an example for strict sense stationary random process.

8. Prove that a first order stationary random process has a constant mean.
Proof:

f x(t )  f x(t  h) as the process is stationary.

E[ X (t )]   x(t ) f x(t )dt

E[ X (t  h)]   x(t  h) f x(t  h)d (t  h)

Put t  h  u  d ( t  h )  du

 x (u ) f  x (u ) du
 E  X (u ) 
 E [ X ( t  h )]  E [ X ( t )]

Therefore, E[ X (t )] is independent of t.

 E[ X (t )] is a constant.

9. What is a Markov process?

Solution:
Markov process is one in which the future value is independent of the past values, given the
present value.

10. Give an example of a Markov process.

Solution:
Poisson process is a Markov process. Therefore, number of arrivals in (0,t) is a Poisson process
and hence a Markov process.

11. Define Markov chain and one – step transition probability.

Solution:

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X  an X  an
If  n, P  n X  a n  2 ,...... X 0  a 0   P  n  then the
 X n 1  a n 1 , n  2   X n 1  a n 1 
process X n  , n=0,1,2,…. is called a Markov chain.

X  a j 
The conditional probability P  n  is called the one step transition probability
 X n 1  a i

from state ai to state a j at the n th step.

12. Describe a random walk process. Is it a Markov process?

Solution:
Suppose a person tosses a fair coin every T seconds and instantly after each toss, he moves a
distance d to the right if heads show and to the left if tails show. X (nT )  is the position of the
person after n tosses

Then the process X (nT ) is a random walk process. The random walk process is a Markov
process.

13. Define Poisson process.

Solution:
If X (t ) represents the number of occurrences of a certain event in (0,t), then the discrete
process X (t ) is called the Poisson process.

14. What is homogeneous Poisson process?

Solution:
e  t (t ) n
The probability law for the Poisson process is P X (t )  n  , n  0,1,2,..... when  is
n!
a constant, the Poisson process is called a homogeneous Poisson process.

15. State the postulates of Poisson process.

The postulates of Poisson process are

(i) P0 occurencein (t , t  t )  1  t  O(t )

(ii) P2 or more occurencesin (t , t  t )  O(t )

(iii) X (t ) is independent of the number of occurrences of the event in any interval prior
to and after the interval (0,t).
(iv) The probability that the event occurs a specified number of times in
(t 0 , t 0  t ) depends only on t , but not on t 0
16. State any two properties of Poisson process.

(i) The Poisson process is a Markov process.

(ii) Sum of two independent Poisson processes is a Poisson process.

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(iii) Difference of two independent Poisson processes is not a Poisson process.

17. Prove that the sum of two independent Poisson processes is also Poisson.
Solution:
Let X (t )  X 1 (t )  X 2 (t )

PX (t )  n  PX 1 (t )  X 2 (t )  n
n
  PX 1 (t )  r PX 2 (t )  n  r 
r 0
n
e 1t (1t ) r e 2t (2t ) n  r

r 0 r! (n  r )!
n
1r t r n2 r t n  r
 e 1t e 2t 
r  0 r!( n  r )!
n
nCr n r n  r
 e ( 1  2 )t  t 1 2
r  0 n!

 e ( 1  2 )t
tn n
n!

2  nC1n211  nC2 n2 2 12  .........1n 
tn
 e ( 1  2 )t (1  2 ) n
n!
tn
 e ( 1  2 )t (1  2 ) n
n!

(    )t ((1   2 )t ) n
=e 1 2
n!

Therefore, X 1 (t )  X 2 (t ) is a Poisson process with parameter (1  2 )t .

18. Prove that the difference of two independent Poisson processes is not a Poisson process.
Solution:
Let X (t )  X 1 (t )  X 2 (t )

E  X (t )  E  X1(t )  X 2 (t )  E  X1(t )  E  X 2 (t )  1t  2t  t (1  2 )

  
E X 2 (t )  E  X 1 (t )  X 2 (t ) 
2

19. Let X (t ) be a Poisson process with rate  . Find E X (t ) X (t   ) (or) Derive the auto
correlation of the poisson process (May/ June 2016)
Solution:

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E  X ( t ) X ( t   )   E  X ( t )  X ( t   )  X ( t )  X ( t )  

 E  X ( t )  X ( t   )  X ( t )    E  X 2 ( t ) 
 

 E  X (t )  E  X (t   )  X (t )   E  X 2 (t )   E  X (t )  E  X ( )   E  X 2 (t ) 
   

 t  t   2t 2   2t   2t 2  t   2t (t   )  t

 E X (t ) X (t   )   2 t (t   )  t

20. For a Poisson process with parameter  and for st show that
k nk
s  s
PN ( s )  k N (t )  n   nC k   1   , k  0,1,2,...n
t  t
Proof:
P   N s   k   N (t )  n   P   N s   k   N (t  s)  n  k  
P  N (s)  k N (t )  n    
P  N (t )  n  P  N (t )  n 
e s ( s)k e (t  s) ( (t  s))n  k
P   N s   k  P  N (t  s)  n  k  k! (n  k )!
  
P  N (t )  n  e  t (t )n
n!
n! e   s k k
 s e e   t  s n  k (t  s) n  k

k !(n  k )! et  nt n
nk nk
 nsk t n  k 1  s  sk t nt k 1  s 
 nCk  t  nCk  t
 tn n t n
k nk
 nCk s 1  s 
tk  t 

k nk
 P  N (s)  k N (t )  n   nCk  s  1  s  , k  0,1,2,....n
t  t

21. For the sine wave process X (t )  Y cos t ,    t  ;   constant, the amplitude Y is a
random variable with uniform distribution in the interval 0 to 1. Check whether the process is
stationary or not.
Solution:
Given Y is a random variable with uniform distribution in the interval 0 to 1, then
f Y ( y)  1 0  y 1

Also given X (t )  Y cos t

E  X (t )   E Y cost   costE  1  cost  0  1   1 cost

 
 2  2

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Since the mean is time dependent, the process is not stationary.

 0 1 
22. If the TPM of a Markov chain is  1 
1  , find the steady state distribution of the chain.

 2 2
(May/ June 2013).

Solution:
 0 1 
 1  2  1
 1    1  2 

 2 2
 1   2   1
1
 1 
3
2
2 
3

23. If N(t) is the poisson process, then what can you say about the time we will wait for the first
event to occur? And the time we will wait for the nth event to occur? (May/ June 2013)
Solution:
e  t (  t ) n
PN (t )  n  , n  0,1,2,.....
n!

Where N(t) is the number of occurrences in (0, t)

24. Is poisson process stationary? Justify? (May/ June 2013) (April/ May 2015)

Solution:

It is a non-stationary process. E ( X (t ))  t is the mean of the poisson process, which depends

on time.

25. Prove that the first order stationary random process has a constant mean.(Nov./Dec. 2013)

E ( X (t   ))  x

f ( x, t   ) dx


 x

f ( x, t ) dx  E[ X (t )]

26. Prove that Poisson process is a Markov process. (Nov./Dec.2013)

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P  X (t3 )  n3 / X (t2 )  n2 , X (t1 )  n1  

P  X (t3 )  n3 , X (t2 )  n2 , X (t1 )  n1 
P  X (t2 )  n2 , X (t1 )  n1 
e   ( t3 t2 )   (t3  t2 )  3
n  n2


(n3  n2 )!
 P  X (t3 )  n3 / X (t2 )  n2 

27. A gambler has Rs.2. He bets Re.1 at a time and wins Re.1 with probability ½. He stops playing
if he loses Rs.2 or wins Rs. 4. What is the transition probability matrix of the related markov
chain? (May/ June 2014)

1 0 0 0 0 0 0
1 
 2 0 1
2 0 0 0 0
0 1
2 0 1
2 0 0 0
 
0 0 0 0 0
1 1
2 2
0 0 0 1 0 1 0
 
2 2

0 0 0 0 1
2 0 1
2
0 1 
 0 0 0 0 0

28. What is a random process? When do you say a random process is a random variable? (April/
May 2015)

A random process is a collection of random variables indexed by the time set T , i.e. X (t , s ) ,

When t is fixed, X(t,s) will become X(s) – a random variable.

29. A radioactive source emits particles at a rate of 5 per min. in accordance with Poisson process.
Each particle emitted has a probability 0.6 of being recorded. Find the probability that 10
particles are recorded in 4 min period. (Nov./ Dec. 2014)

e  (5(0.6)(4) (5(0.6)(4))10
P ( X (4)  10) 
10!
12 10
e (12)

10!
 0 1 0 
 
30. Check whether the Markov chain with transition probability matrix P   1 0 1  is
2 2
 
 0 1 0 
irreducible or not? (Nov./Dec.2014)

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Given TPM is an irreducible matrix, because each state is reached from the remaining states in
some non-zero number of steps.

1 2

31. The random process X(t) is given by X (t )  Y cos(2 t ), t  0 Where Y is a random variable
with E(Y)=1. Is the process X(t) stationary? (May/ June 2016)

X (t )  Y cos(2 t ), t  0
E ( X (t ))  E (Y ) cos(2 t )  cos(2 t )
which depends on ' t '

X(t) is not a stationary process

32. When a Markov chain is called homogeneous? (Nov./Dec. 2015)

33. Consider a random process X (t )  cos t ,    t  ; where  is a constant and  is uniform

 
variable in  0,  , show that X(t) is not WSS? (Nov./Dec. 2015)
 2

 0 1 1 
 2 2
34. Consider a markov chain with state {0, 1, 2} and TPM P   1 0 1  , Draw the
 2 2
 1 0 0 
 
transition diagram. (Nov. Dec. 2015)

PART-B

1. If customers arrive at a counter in accordance with a Poisson process with a mean rate of 2/min,
find the probability that the interval between 2 consecutive arrivals is more than 1 min, between
1 and 2 mins and 4 mins or less. (AP) (Nov/Dec 2010, 2011)
2. At the receiver of an AM radio, the received signal contains a cosine carrier signal at the carrier
frequency  0 with a random phase  that is uniformly distributed over (0,2 ) . The received
carrier signal is X (t )  A cos( o t   ) . Show that the process is second order stationary. (AP)
(May/Jun 2007)
3. Show that the random process X (t )  A cos( 0 t   ) is wide sense stationary if A and  0 are
constants and  is uniformly distributed random variable in (0,2 ) . (Nov/Dec 2007,2011)
(Nov./Dec.2015) (AP)

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4. Show that random process X (t )  A cos t  B sin t ,    t   is a wide sense stationary

process where A and B are independent random variables each of which has a value -2 with
probability 1/3 and a value 1 with probability 2/3. (April/May2011)(April/May2015)
(Nov./Dec.2015) (AP)
5. Given a RV  with density f ( ) and another RV  uniformly distributed in (  ,  ) and
independent of  and X (t )  a cos( t   ) , prove that X (t ) is a WSS process. (AP)
(May/Jun 2006) (Nov/Dec 2008)
6. Consider a random process X (t ) defined by X (t )  U cos t  V sin t where U and V are
independent random variables for which E (U )  E (V )  0; E (U 2 )  E (V 2 )  1
(1) Find the auto covariance function X t 
(2) Is X t  wide sense stationary? Explain your answer. (U)(Apr/May 2008)
7. A stochastic process is described by X (t )  A sin t  B cos t where A and B are independent
random variables with zero means and equal standard deviation show that the process is
stationary of the second order. (AP)(Nov/Dec 2006)
8. Suppose that a mouse is moving inside the maze shown in the adjacent figure from one cell to
another, in search of food. When at a cell, the mouse will move to one of the adjoining cells
randomly, for n  0 , X n be the cell number the mouse will visit after having changed cells
‘n’ times. Is X n ; n  0,1,... a Markov chain? If so, write its state space and transition
probability matrix. (AP)
1 4 7

2 5 8

3 6 9

9. A raining process is considered as two states Markov Chain. If it rains, it is considered to be

state 0 and if it does not rain, the chain is in state 1. The transition probability of the Markov
0.6 0.4
chain is defined as P    Find the probability that it will rain for 3 days from today
0.2 0.8
assuming that it will rain after 3 days. Assume the initial probabilities of state 0 and state 1 as
0.4 and 0.6 respectively. (AP)(Nov/Dec 2006)
 1 1
0 0 2 2 
 
10. Let the Markov chain consisting of the states 0,1,2,3 have the tpm 1 0 0 0  . Determine
0 1 0 0 
0 1 0 0 
 
which states are transient and which are recurrent by defining transient and recurrent states.
(AP) (April/May 2010)

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11. The following is the transition probability matrix of a Markov chain with state space
0,1,2,3,4. Specify the classes are transient and which are recurrent. Give reasons.
2 / 5 0 0 3/ 5 0 
1 / 3 1 / 3 0 1 / 3 0 

 0 0 1/ 2 0 1/ 2 . (AP) (Nov/Dec 2010)
 
1 / 4 0 0 3/ 4 0 
 0 0 1 / 3 0 2 / 3

12. Suppose that whether or not it rains today depends on previous weather conditions through the
lasts two days. Show how this system may be analyzed using a Markov chain. How many
stats are needed? (AP)(April/May 2010)
13. A raining process is considered as two states Markov Chain. If it rains, it is considered to be
state 0 and if it does not rain, the chain is in state 1. The transition probability of the Markov
0.6 0.4
chain is defined as P    Find the probability that it will rain for 3 days from today
0.2 0.8
assuming that it is raining today. Find also the unconditional probability that it will rain after
three days with the initial probabilities of state 0 and state 1 as 0.4 and 0.6 respectively
(AP) (May/Jun 2006)
14. A person owning a scooter has the option to switch over to scooter, bike or car next time with
 0.4 0.3 0.3
the probability of (0.3, 0.5, 0.2). If the transition probability matrix is  0.2 0.5 0.3 .
0.25 0.25 0.5
What are the probabilities vehicles related to his fourth purchase? (AP)(Nov/Dec 2006)
15. Find the limiting-state probabilities associated with the following transition probability matrix
0.4 0.5 0.1
 0 .3 0 .3 0 .4  (AP)(April/May 2011)
 
0.3 0.2 0.5
16. An engineer analyzing a series of digital signals generated by a testing system observes that
only 1 out of 15 highly distorted signals follows a highly distorted signal, with no
recognizable signal between, whereas 20 out of 23 recognizable signals follow recognizable
signals, with not highly distorted signal between. Given that only highly distorted signals are
not recognizable, find the TPM and fraction of signals that are highly distorted. (AP)
(May/Jun 2009) (Nov/Dec 2007) (Nov/Dec 2010)(Nov./Dec.2014)(April/ May 2015)
17. A salesman territory consists of three cities A, B and C. He never sells in the same city on
successive days. If he sells in city-A, then the next day he sells in city-B. However if he sells
in either city-B or city-C, the next day he is twice as likely to sell in city-A as in the other
city. In the long run how often does he sell in each of the cities? (AP)(April/May 2012)
(Nov/Dec.2013)
18. Derive Chapman-Kolmogorov equations. (AN)(April/May 2010)(April/ May 2015)

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19. Derive probability distribution of poisson process and hence find its auto correlation function.
(AN)(April/May 2011)
20. Show that the difference of two independent poisson processes is not a poisson process. (AN)
(April/May 2011)(May/ June 2013)
21. Define Poisson process and derive the Poisson probability law. (AN)(Nov/Dec 2011)
22. Three out of every four trucks on the road are followed by a car, while only one out of every
five cars is followed by a truck. What fraction of vehicles on the road are trucks? (AP)
(April/May2010)
23. The process X (t ) whose probability distribution is given by
 ( at ) n 1
 , n  1,2,...
PX (t )  n  
(1  at ) n 1 Show that X (t ) is not stationary. (May/Jun2006)
 at , n  0.
1  at
(Apr/May 2012)(Nov./Dec.2013) (Nov./Dec.2015) (AP)
(1) The transition probability matrix of a Markov chain X n , n  1,2,3.... having 3 states
 0 .1 0 .5 0 .4 
1, 2 and 3 is 0.6 0.2 0.2 and the initial distribution is 0.7,0.2,0.1 . Find PX 2  3 ,
0.3 0.4 0.3
P  X 3  2, X 2  3, X 1  3 X 0  2 . (Nov/Dec2008)(Nov./Dec.2013)(May/June2014)
(Nov./Dec.2015) (AP)
24. Assume that a computer system is in any one of the three states: busy, idle and under repair
respectively denoted by 0, 1, 2. Observing its state at 2 pm each day, we get the transition
0.6 0.2 0.2
probability matrix as P   0.1 0.8 0.1 . Find out the 3rd step transition probability matrix.
0.6 0 0.4
Determine the limiting possibilities. (AP)(May/Jun 2007)
25. Define stationary transition probabilities. Derive the Chapman- Kolmogorov equations for
discrete time Markov chain. (AN)(Nov/Dec 2007)
26. On a given day, a retired English professor Dr. Charles Fish, amuses himself with only one of
the following activities: reading (activity 1), gardening (activity 2), or working on his book
about a river valley (activity 3). For 1  i  3 let X n  i if Dr. Fish devotes day ‘n’ to activity
i. Suppose that X n , n  1,2,3.... is a Markov chain and depending on which of these
0.30 0.25 0.45
activities on the next day is given by TPM 0.40 0.10 0.50 . Find the proportion of days
0.25 0.40 0.35
Dr. Fish devotes to each activity. (AP)(Nov/Dec 2007)
27. Suppose that customers arrive at a bank according to a Poisson process with a mean rate of 3
per minute; find the probability that during a time interval of 2 minutes
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(1) exactly 4 customers arrive (2) more than 4 customers arrive (3) Fewer than 4
customers arrive (AP) (May/Jun 2006)(Nov./Dec.2013)(April/ May 2015) (Nov./Dec.2015)
28. Queries presented in a computer database are following a Poisson process of rate   6 queries
per minute. An experiment consists of monitoring the database for m minutes and recording
N(m) the number of queries presented.
a) What is the probability that there are no queries in a one minute interval?
b) What is the probability that exactly 6 queries arrive in a one minute interval?
c) What is the probability of less than 3 queries arriving in a half minute interval?
(AP)(May/Jun 2007).
29. Obtain the steady state or long run probabilities for the population size of a birth death process.
(AN)(May/Jun 2007)
30. Discuss the pure birth process and hence obtain its probabilities, mean and variance.
(AN)(Apr/May 2008)
31. Write a short note on recurrent state, transient state, ergodic state. (R)(Nov/Dec 2008)
 
32. Let X (t ) be a Poisson process with arrival rate  . Find E  X (t )  X ( s)  t  s . (AN)
2

(Apr/May 2008)
33. Let X n ; n  1,2,3,... be a Markov chain on the space S  1,2,3with one step transition
0 1 0
1 1
probability matrix P   0 
 12 0 02 
 
(a) Sketch the transition diagram.
(b) Is the chain irreducible? Explain
(c) Is the chain ergodic? Explain. (AP)(Apr/May 2008)(May/ June 2013)
34. A man either drives a car or catches a train to go to office each day. He never goes 2 days in a
row by train but if he drives one day, then the next day he is just as likely to drive again as he is
to travel by train. Now suppose that on the first day of the week, the man tossed a fair die and
drove to work iff a 6 appeared. Find (1) the probability that he takes a train on the 3 rd day, (2)
the probability that he drives to work in the long run. (Nov/Dec 2008) (Nov/Dec 2011)
(Nov./Dec.2015) (AP)
35. Show that the process X (t )  A cos t  B sin t is wide sense stationary, if
E ( A)  E ( B )  0, E ( A 2 )  E ( B 2 ) and E ( AB )  0 where A and B are random variables.
(AP)(May/ June 2013)(April/May 2015)
36. Prove that the poisson process is a Markov process. (AN)(May/ June 2013)

37. A fair die is tossed repeatedly. The maximum of the first ‘n’ outcomes is denoted by X n . Is
X n , n  1,2,.... a Markov chain why or why not? IF it is a Markov chain, calculate its
transition probability matrix. Specify the classes. (AP) (Nov./Dec.2012)(April/May 2015)

38. An observer at a lake notices that when fish are caught, only 1 out of 9 trout is caught after
another trout, with no other fish between whereas 10 out of 11 non-trout, with no other fish
between whereas 10 out of 11 non-trout are caught following non-trout, with no trout
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between. Assuming that all fish are equally likely to be caught, what fraction of fish in the
lake is trout? (AP)(Nov./Dec.2012)(April/ May 2015)
39. The following is the transition probability matrix of a Markov chain with state space
1,2,3,4,5. Specify the classes are transient and which are recurrent. Give reasons. (AP)
0 0 0 0 1 
0 1 / 3 0 2 / 3 0 
 
0 0 1 / 2 0 1/ 2 . (Nov/Dec 2012)
 
0 0 0 1 0 
0 0 2 / 5 0 3 / 5
40. For an English course, there are four popular textbooks dominating the market. The englush
department of an institution allows its faculty to teach only from these 4 text books. Each
year, prof. Rose Mary O donoghue adopts the same book she was using the previous year
with probability 0.64. The probabilities of her changing to any of the other 3 books are equal.
Find the proportion of years Prof. O’ Donoghue uses each book. (AP)(Nov./ Dec.2012)

41. Explain the steady state probabilities of birth-death process. Also draw the transition graph?
(U)(Nov./Dec. 2012)

42. Show that the sum of two independent poisson process with parameter 1 and 2 is also a
poisson process (U)(May/ June 2014)

43. Find the limiting-state probabilities associated with the following transition probability matrix
 0.5 0.4 0.1
 0.3 0.4 0.3
 
 0.2 0.3 0.5 

45. A soft water plant works properly most of the time. After a day in which the plant is working,
the plant is working the next day with probability 0.95. Otherwise a day or repair followed by
a day of testing is required to restore the plant to working status. Draw the state transition
diagram for the status of the plant. Write down the TPM and Classify the status of the
process. (AP)(Nov./Dec.2014)
46. Suppose that children are born at a poisson rate of five per day in a certain hospital. What is
the probability that (1) atleast two babies are born during the next six hours. (2) no babies are
born during the next two days? (AP)(Nov./Dec.2014)
47. If {N1(t)}and {N2(t)}are two independent Poisson process with parameter 1 and
n
2 respectively, show that P ( N1 (t )  k / N1 (t )  N 2 (t )  n)    p k q n  k , where
k 
1 1
p and q  . (AN)(Nov./Dec.2015)
1  2 1  2

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48. Consider a random process Y (t )  X (t ) cos( w0t   ) where X(t) is WSS process,  is a
uniformly distributed r.v. over   ,   w0 is a constant. It is assumed that X(t) and  are
independent. Show that Y(t) is WSS? (AP)(May/ June 2016)
 0.4 0.6 
49. Consider the Markov chain Xn, n=0,1,2.. having state space and TPM P=   (1)
 0.8 0.2 
Draw the transition diagram (2) IS the chain irreducible? (3) Is the state -1 ergodic? Explain
(4) IS the chain ergodic? Explain (AN)(May/ June 2016)
50. Let X(t) and Y(t) be two independent poisson process with parameters 1 and 2 resp. Find
(1) P(X(t)+Y(t)) = n, n=0,1,2,… (2) P(X(t)-Y(t)) = n, n=0, 1,-1,2,-2,…. (AN) (May/ June
2016)
51. Let X n ; n  1,2,3,... be a Markov chain on the space S  1,2,3with one step transition
0 1 0
1 1
P
2 
probability matrix 0 with initial state probability
2
1 0
 0
i
distribution P ( X 0  i )  , i  1, 2,3 . Find (1) P ( X 3  2, X 2  1, X 1  2 / X 0  1)
3
(2) P ( X 3  2, X 2  1, X 1  2 , X 0  1) , (3) P ( X 2  2 / X 0  2) ,
(4) Invariant probabilities of the Markov chain. (AP)(May/ June 2016)

COURSE OUTCOME: Able to characterize phenomena which evolve with respect to time in
probabilistic manner. Choose an appropriate method to solve a practical problem.

UNIT IV

MARKOVIAN QUEUEING MODELS

COURSE OBJECTIVE: Ability to analyze basic properties of Markov chains and their applications
in modeling queuing systems and acquire skills in analyzing queueing models. Have an ability to
design a model or a process to meet desired needs within realistic constraints such as environmental
conditions

FORMULAS

MODEL-I: ( M / M / 1) : ( / FIFO )

1. Steady state probability p0


p0  1 


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n n
   
p n    , p0    1  
   

2. Average no. of customers in the system Ls


Ls 


3. Average number of customers in the queue ( Lq )

Lq  Ls        2
       

4. Average waiting time of a customer in the system Ws

Ls  1
Ws   
         

5. Average waiting time of a customer in the system Ws

Lq 2 
Wq   
           

6. Probability that the number of customers in the system exceeds ‘k’

k 1

p( N  k )   


7. Average number Lw of customers in the non-empty queue ( Lw )


Lw 
 

8. Average waiting time of customers in the queue ,if he has to wait.

E Wq Wq  0  
1
 

9. Probability density function (pdf) of the waiting time in the system

f (w)  (   ) e(   )w

10. Probability that the waiting time of a customer in the system exceeds ‘t’

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 (   ) w
  (   ) e , w0

g ( w)  
1   , w0
 

NOTE: Wq does not follow an exponential distribution.

MODEL-II: ( M / M / c ) : ( / FIFO )

   n
  p 0 , ns
  
1. pn   n
 1 
 
 s !s ns    p0 , ns
  
s 1

 
2. Lq 
1  p0
2
s !s   
1  
  s
s 1

 
3. L s  Lq 


1  p0 

 s !s   
2

1  
  s
  s
 
4. Ws 
Ls 1
 
1 1  p0
   s ! s 2
  
1  
  s
s

 
5. Wq 
Lq 1 1
 p0
  s !s   
2

1  
  s
1
6. p0 
s 1
1  n
 s   

n 0 n ! 
n
  s 1 
  s! 
 
  s 

MODEL-III ( M / M / 1) : ( / FIFO )

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 

1  
  n   
1. pn      , if   
     k 1 
1   
    
   

pn  1 , if   
k 1

1 
p0   if   
  k 1
1   
 

p0  1 , if   
k 1

2. Ls  The average no. of customer s in the system

Ls  k , if   
2

 k 1 
 (k  1)    
 
       
Ls    , if   
      k  1 
 1    
   

3. Lq  Ls    where   is the effective arrival rate     (1  p 0 )



Ls
4. W s 


Lq
5. Wq 


MODEL-IV ( M / M / s ) : ( k / FIFO )

 n
1  ns

 n!    p
0
,


1.  1   n
p    p , snk
s! s n  s   
n  0

0 , nk




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
2.   s   k  s  (k  s)(1   )  k  s  where  
Lq  p  
0 1    
s ! 1   2  

  s 1  k
Ls  Lq   where      s   (s  n) pn    pn  1
3. 
 n 0  n 0

4. Ws  Ls


Lq
5. Wq  where   is the effective arrival rate

1
6. p 
0 s 1  
  n  s 
k
n  s
 
1
    
n!
n  s s !  s 
n 0  n  s 

PART-A

1. In a given M / M / 1 /  / FCFS queue   0.6 , what is the probability that the queue contains 5
or more customers?
5

Solution: p (n  5)      5  (0.6) 5

2. What is the probability that a customer has to wait more than 15 min. to get his service
completed in ( M / M / 1) : ( / FIFO ) queue system if   6 / min . &   10 / hr ? (Nov/ Dec.
2012)
1
Solution: p( ws  15 min)  p( ws  hr ) , ws is exp onential with parameter   
4

  (    ) e (   ) d
1
4
(   )  (10  6 )

  e  (    ) 

1
4
 0e 4
e 4
 e 1

3. A duplicating machine maintained for office use is operated by an office assistant. If jobs arrive
at a rate of 5 per hour and the time to complete each job varies according to an exponential
distribution with mean 6 min., find the percentage of idle time of the machine in a day.
(Assume that jobs arrive according to a poisson process)
Solution: ( M / M / 1) : ( / FIFO ) model

1
  5 / hr ,   min 10 / hr
6

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 5
p0  1   1  0 .5
 10

50 % of time the machine is idle.

4. For ( M / M / 1) : ( / FIFO ) model write down the little’s formula.

Solution: Ls   Lq , Ws   Wq , Ls  ws   , Ws  Wq  
 

5. Consider an (M/M/1) queueing system. If   6 and   6 find the probability of atleast 10

customers in the system.
10

10
6
p (n  10)        0.75
10
Solution:
 8
6. What are the basic characteristics of Queueing process?

Solution:

1. Arrival time pattern (distribution) (a)

2. Service time pattern(distribution) (b)
3. No. of services (c)
4. Capacity of the system (d)
5. Service discipline (Queue discipline) (e)

7. Write the Kendall’s notation and explain.

Solution:
( a / b / c ) : ( d / e) is the Kendall’s notation a , b , c , d , e are explained as above in Q.no.(6)
8. Derive the average no. of customers in the system for ( M / M / 1) : ( / FIFO )
Solution:

Ls   n pn  0. p0 1. p1  2. p2  3. p3  .....
n0
 n  2 3 
  n    p  p    2    3    ....
       

n1     0 0 

 

  
 
 2 
 p0      
 1 2    3    ....
    
 
 

 p0 

1
2
1 
 
 
  


 1 
   1  
.
   2  
1  
   


 
9. What is the probability that an arrival to an infinite capacity 3 server poisson queue with
 2 1
 and p0  enters the service without waiting?
c 3 9
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Solution:
Arriving customer shall enter the system without waiting if number of customer in the
system  Number of server (=3)

 p (n  3)  p 0  p1  p 2 n
1 
1 1 
2
 p n    p 0
 p0  p 0    p 0 n!   
1 2 

 2  2 
Given      2 and c  3
c 3 3 3 

1 1 1 1 5
 p(n  3)   2.  4 
9 9 2 9 9

10. Consider an M/M/C queueing system. Find the probability that an arriving customer is
forced to join the queue.

Solution: Arriving customers has to join the queue if

Number of customers in the system  number of server = c

p (n  c)  1   p 0  p1  p 2  ....  p c 1 
 1 1
2
1
3
1 
c 1

1   p0  p 0    p 0    p 0  .....    p 0 
 1!  2!    3!    (c  1)!    
 1  1    2 1   3 1 
c 1

 1  p 0 1         .....    
 1!  2!    3!    (c  1)!    

11. What is the probability that an arrival to an infinite capacity 3 server poisson Queueing
 1
system with  2 and p0  enters the service without waiting?
 9

Solution: ( M / M / 3) model

Customers enter the system without waiting if less than 3 customers in the system
2
p(n  3)  p  p  p  p  1  p  1    p  1  2  1 4 1  5 .
0 1 2 0 1!  0 2    0 9 9 2 9 9

12. What is the effective arrival rate for ( M / M / 1 / 4 / FCFS ) queueing model?
Solution:     (1  p 0 )

13. For ( M / M / c ) : ( N / FIFO ) model , write down the formula for

(a) Average no. of customers in the system (b) average waiting time in the system

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Solution:

(a) Ls  Lq  

s
where Lq  p  
 

 
 1  k s  (k  s)(1  ) k s 

0    2 
s! 1   

 s1 
     s   (s 1) pn  and   s

 n0 


L
(b) Ws  s


14. What are the characteristics of a queuing system? (May/ June 2013)

Solution:
The basic characteristics of a queueing system are

(a) Arrival pattern (b) Service pattern (C) Number of servers

(b) Number of servers

(c) System Capacity

(d) Queue discipline

15. What is the probability that a customer has to wait more than 15 minutes to get his service
completed in a M/M/1 queueing system, if   6 per hour and   10 per hour?
(May/ June 2013)
Solution:
 (   )t
Probability that the waiting time in the queue exceeds t  e

6 (106 )15 6 60
Probability that the waiting time exceeds 15 minutes  e  e  5.254  10  27
10 10
16. Give a real life situation in which (a) customers are considered for service with last in first
out queue discipline (b) a system with infinite number of servers. (Nov. / Dec. 2012)

a. Inventory systems
b. A super market bill counter, Telephone exchange long distance operators, Petrol pump

17. Consider a random queue with two independent Markovian servers. The situation at server 1
is just as in M/M/1 model. What will be the type of queue in server 2? Why?(Nov/ Dec.
2012)

Solution:
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The type of queue in server 2 is FIFO with infinite capacity.

18. Define Markovian Queueing models. (Nov./Dec.2013)

Queueing models in which both inter arrival time and service time which are exponentially
distributed are called Markovian Queueing models.

19. Suppose that customers arrive at a poisson rate of one per every 12 minutes and that the
service time is exponential at a rate of one service per 8 minutes. (a) what is the average
number of customers in the system? (b) What is the average time of a customer spends in the
system? (Nov./Dec.2013)

1 1
 / min;   / min
12 8
 2
 
 3

The average number of customers in the system is Ls  2
1 
1
The average time the customer spends in the system Ws  Ls  24 min

20. A supermarket has a single cashier. During peak hours, customers arrive at a rate of 20 per
hour. The average number of customers that can be serviced by the cashier is 24 per hour.
Calculate the probability that the cashier is idle. (May,/June, 2014)

20 20 1
 , P0  1    0.1667
24 24 6
21. State the steady state probabilities of the finite source queueing model represented by (M/M/R):
(GD/K/K) (May/ June ,2014)

1
P0  n s ns
c 1
1  1 k
1  

n0
 
n!  
  
s!  

ns n !  s  
1  n

   P0 , n  s
 n!  
 n
 1 
Pn   ns   P0 , s  n  k
 s !s   
0 , n  k




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22. State the relationship between expected number of customers in the queue and in the system
(Nov./Dec.2014)

Expected number of customer in the system is Ls

Expected number of customer in the queue is Lq


Ls  Lq 


23. What is the steady state condition for M/M/C queueing model? (Nov./Dec.2014)(May/ June
2016)

For Multi server , infinite capacity model:

   n
  p 0 , ns
  
pn   n
 1 
 s ! s n  s    p 0 , ns
  

1
p0 
s 1
1  n
 s   

n 0 n ! 
n
  s 1 
  s! 
 
  s 

For multi server, finite capacity model:

 n
1  ns

 n!     p
0
,


p  1   n
snk
n  s   
n  p ,
0
 s ! s


0 , nk

1
p 
0 s 1  
  n  s 
k
n  s
 n! 
1
    
n  s s !  s 
n 0  n s 

24. What do the letters in the symbolic representation (a / b / c)(d / e) of a queueing model
represent (April/ May 2015)

Arrival time pattern (distribution) (a)

Service time pattern(distribution) (b)
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No. of services (c)

Capacity of the system (d)
Service discipline (Queue discipline) (e)

25. What do you mean by balking and reneging? (April/ May 2015)(May/ June 2016)

Balking: A customer may decide to wait no matter how long the queue becomes, or if he
queue is too long to suit him, may decide not to enter it. If a customer decides not to enter the
queue upon arrive, he is said to have balked.

Reneging: Sometimes a customer may enter the queue, but after he may decide to leave the
queue due to impatience, In this case he is said to have reneged.

26. Draw the state transition rate diagram of a M/M/C queueing model (April/ May 2015)
 2
1
S1 S2


27. What is the probability that a customer has to wait more than 15 min. to get his service
completed in a M/M/1 queueing system, if   6 per hour and   10 per hour? (April/ May
2015)

P (W  t )  e  (   )t
15
15  10  6 
P (W  )  e 60
 e 1
60

28. What effect does doubling  and  have on Ls and Ws for an

( M / M / 1) : ( FIFO  /  ) queueing model? (Nov./Dec.2015)

29. Write the steady state probabilities for the ( M / M / R ) : (GD / K / K ) , R  K queueing model.
(Nov./Dec.2015)

30. Which queue is called to be the queue with discouragement? (Nov./Dec.2015)

31. What is the effective arrival rate for ( M / M / 1) : ( 4 / FCFS ) queueing model?

(Nov./Dec.2015)

PART-B

1. There are three typists in an office. Each typist can type an average of 6 letters per hour. If
letters arrive for being typed at the rate of 15 letters per hour
(a) What is the probability that no letters are there in the system?
(b) What is the probability that all the typists are busy? (AP)(May/Jun 2007)
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2. Explain an M/M/1, finite capacity queuing model and obtain expressions for the steady state
probabilities for the system size. Find also the mean number of customers in the system.
(AN) (May/Jun 2007)(Apr/May 2008)
n

3. For the steady state M/M/1 queueing model prove that Pn    P0 (AN)(Nov/Dec 2007)

4. Define Kendall’s notation. What are the assumptions made for the simplest queueing model?
(R)
5. Calculate any four measures of effectiveness of M/M/1 queueing system. (AN)(April/May
2010)
6. Derive the formula for the average number of customers in the queue and the probability that
an arrival has to wait for (M/M/C) with infinite capacity. Also derive for the same model the
average waiting time of a customer in the queue as well as in the system. (AN)(May/Jun
2006)
7. Define birth and death process. Obtain its steady state probabilities. How it could be used to
find the steady state solution for the M/M/1 queueing system. Why is it called geometric?
(AN)(April/May 2010)(April/ May 2015)
8. On every Sunday morning, a Dental hospital renders free dental service to patients. As per
hospital rules, 3 dentists who are equally qualified and experienced will be on duty then. It
takes on an average 10 minutes for a patient to get treatment and the actual time taken is
known to vary approximately exponentially around this average. The patients arrive
according to the Poisson distribution with an average of 12 per hour. The hospital
management wants to investigate the following:
(1) The expected number of patients waiting in the queue
(2) The average time that a patient spends at the hospital. (AP)(Nov/Dec 2007)
9. A concentrator receives messages from a group of terminals and transmits them over a single
transmission line. Suppose that messages arrive according to a Poisson process at a rate of
one message every 4 milliseconds and suppose that message transmission times are
exponentially distributed with mean 3 minutes. Find the number of messages in the system
and the mean total delay in the system. What percentage increase in arrival rate results in a
doubling of the above mean total delay? (AP)(Apr/May 2008)
10. A duplicating machine maintained for office use is operated by an office assistant who earns
Rs.5 per hour. The time to complete each job varies according to an exponential distribution
with mean 6 minutes. Assume a Poisson input with an average arrival rate of 5 jobs per hour.
If an 8 – hrs day is used as a base, determine
(a) The percentage idle time of the machine.
(b) The average time a job is in the system.
(c) The average earning per day of the assistant. (AP)(Nov/Dec 2008)
11. Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per hour. The
waiting room does not accommodate more than 14 patients. Examination time per patient is
exponential with mean rate of 20 per hour.
(1) What is the effective arrival rate?
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(2) What is the probability that an arriving patient will not wait?
(3) What is the expected waiting time until a patient is discharged from the clinic?
(AP)(May/Jun 2007) (Nov/Dec 2015)
12. There are three typists in an office. Each typist can type an average of 6 letters per hour. If
letters arrive for being typed at the rate of 15 letters per hour
(a) What fraction of the time all the typists will be busy?
(b) What is the average number of letters waiting to be typed?
(c) What is the average time a letter has to spend for waiting for being typed?
(d) What is the probability that a letter will take longer than 20 minutes waiting to be
typed and being typed?
(AP) (May/Jun 2009)(Nov/Dec 2010,2011)(Apr/May 2012)(May/ June 2013)
13. Self-service system is followed in a super market at a metropolis. The customer arrivals occur
according to a Poisson distribution with an average of 12 per hour. The hospital management
wants to investigate the following:
(a) Find the expected number of customers in the system.
(b) What is the percentage of time that the facility is idle? (AP)(Nov/Dec 2007)
14. A supermarket has two girls attending to sales at the counters. If the service time for each
customer is exponential with mean 4 minutes and if people arrive in Poisson fashion at the
rate of 10 per hour,
(a) What is the probability that a customer has to wait for service?
(b) What is the expected percentage of idle time for each girl? (AP)(Nov/Dec 2008)
(c) What is the expected length of customer’s waiting time?
(AP)(May /June 2012)(April/ May 2015)
15. A T.V. repairman finds that the time spent on his job has an exponential distribution with
mean 30 minutes. If he repair sets in the order in which they came in and if the arrival of sets
is approximately Poisson with an average rate of 10 per 8 hour day. What is the repairman’s
expected idle time each day? How many jobs are ahead of average set just brought?
(May/June 2012) (Nov./Dec.2013) (Nov./Dec.2015) (AP)
16. Customers arrive at one window drive-in bank according to Poisson distribution with mean
10 per hour. Service time per customer is exponential with mean 5 minutes. The space is front
of window, including that for the serviced car can accommodate a maximum of three cars.
Others cars can wait outside this space.
(i) What is the probability that an arriving customer can drive directly to the space in
front of the window?
(ii) What is the probability that an arriving customer will have to wait outside the
indicated space?
(iii) How long is an arriving customer expected to wait before being served?
(AP)(April/May 2011)
17. Customers arrive at a one man barber shop according to a Poisson process with a mean inter
arrival time of 12 minutes. Customers spend an average of 10 minutes in the barber’s chair
(a) What is the expected number of customers in the barber shop and in the queue?
(b) How much time can a customer expect to spend in the barber’s shop?
(c) What is the average time customer spends in the queue?
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(d) What is the probability that the waiting time in the system is greater than 30 minutes?
(AP)(May/June 2009)
18. Customers arrive at a one man barber shop according to a Poisson process with a mean inter
arrival time of 12 minutes. Customers spend an average of 10 minutes in the barber’s chair
(a) What is the expected number of customers in the barber shop and in the queue?
(b) What is the probability that more than 3 customers are in the system? (AP) (Nov/Dec
2008)
19. Find the mean number of customers in the queue, system, average waiting time in the queue
and system of M/M/1 queueing model. (AN)(Nov/Dec 2011)
20. Show that for the ( M / M / 1) : ( FCFS /  /  ) , the distribution of waiting time in the system is
(   )t
w(t )  (    )e , t 0 (AN)(April/May 2011)
21. Find the steady state solution for the multi server M/M/C model and hence find L9, W9, Ws
and Ls by using Little’s formula. (AN)(April/May 2011)
22. If people arrive to purchase cinema tickets at the average rate of 6 per minute, it takes an
average of 7.5 seconds to purchase a ticket. If a person arrives 2 min before the picture starts
and it takes exactly 1.5 min to reach the correct seat after purchasing the ticket.
(i) Can he expect to be seated for the start of the picture?
(ii) What is the probability that he will be seated for the start of the picture?
(iii) How early must he arrive in order to be 99% sure of being seated for the start of the
picture?
(AP)(Nov/Dec2010)(Nov./Dec. 2014)
23. Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line
capacity of the yard is limited to 5 trains, find the probability that the yard is empty and the
average number of trains in the system, given that the inter arrival time and service time are
following exponential distribution. (AP)(April/May 2012)
24. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 minutes
between one arrival and the next. The length of a phone call is distributed exponentially with
mean 4 minutes
(a) What is the average number of customers in the system?
(b) What fraction of the day will the phone be in use?
(c) What is the probability that an arriving customer will have to wait? (AP)(May/Jun
2007)
25. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 minutes
between two consecutive calls arrival. The length of a phone call is distributed exponentially
with mean 4 minutes
(a) Determine the probability that a person arriving at the booth will have ot wait.
(b) Find the average queue length that is formed from time to time.
(c) The telephone company will install a second booth when convinced that an arrival
would be expected to wait at least 5 minutes for the phone. Find the increase in flows
of arrivals which will justify a second booth.
(d) What is the probability that an arrival will have to wait for more than 15 min before
the phone is free? (AP)(Nov/Dec 2006)
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26. A petrol pump has 2 pumps. The service times follow the exponential distribution with mean
of 4 minutes and cars arrive for service is a Poisson process at the rate of 10 cars per hour.
Find the probability that a customer has to wait for service. What is the probability that the
pumps remain idle? (AP) (Apr/May 2008)

27. Automatic car wash facility operates with only one bay. Cars arrive according to a Poisson
process, with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is
busy. If the service time for all cars is constant and equal to 10 minutes, determine

(a) Mean number of customers in the system Ls

(b) Mean number of customers in the queue
(c) Mean waiting time in the system.
(d) Mean waiting time in the queue. (AP) (Apr/May 2008)(May/ June 2013)
26. Derive (1) Ls, average number of customers in the system (2) Lq, the average number of
customers in the queue for the queuing model ( M / M / 1) : ( N / FIFO ) .(May/ June
2013)(May/ June 2014) (AP)
27. Customers arrive a t a one man barber shop according to a poisson process with a mean inter
arrival time of 20 minutes. Customers spend an average of 15 minutes in the barber chair. The
service time is exponentially distributed. If an hour is used as a unit of time, then
(1) What is the probability that a customer need not wait for a hair cut?
(2) What is the expected number of customer in the barber shop and in the queue?
(3) How much time can a customer expect to spend in the barber shop?
(4) Find the average time that a customer spends in the queue?
(5) Estimate the fraction of the day that a customer will be idle?
(6) What is the probability that there will be 6 or more customers?
(7) Estimate the percentage of customers who have to wait prior to getting into the
barber’s chair? (May/Jun 2006)(April/ May 2015) (May/ June 2013) (AP)
28. At a port there are 6 unloading berths and 4 unloading crews/ when all the berths are full,
arriving ships are diverted to an overflow facility 20kms down the river. Tankers arrive
according to poisson process with a mean of 1 every 2 hrs. It takes for an unloading crew, on
the average 10 hrs to unload a tanker, the unloading time following an exponential
distribution. Find (i) how many tankers are at the port on the average? (ii) How long does a
tanker spend at the port on the average? (iii) What is the average arrival rate at the overflow
facility? (AP)(Nov./Dec.2012)
29. Consider a single server Queueing system with poisson input, exponential service times.
Suppose the mean arrival rate is 3 calling units per hour, the expected service time is 0.25
hours and the maximum permissible number calling units in the system is two. Find the
steady state probability distribution of the number of calling units in the system and the
expected number of calling units in the system. (AP)(Nov./Dec.2013)
30. A telephone exchange has two long distance operators. It is observed that, during the peak
load long distance calls arrive in a poisson fashion at an average rate of 15 per hour. The
length of service on these calls is approximately exponential distributed with mean length 5

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minutes. Find (a) the probability a subscriber will have to wait for long distance call during
the peak hours of the day. (b) If the subscriber will wait and are served in turn, what is the
expected waiting time . (AP)(Nov./Dec.2013)
31. Customers arrive at a sales counter manned by a single person according to a poisson process
with a mean rate of 20 per hour. The time required to serve a customer has an exponential
distribution with a mean of 100 seconds. Find the average waiting time of a customer.
(Nov./Dec.2013)(AP)
32. Derive the governing equation for the (M/M/1) : (GD / N/  ) queueing model and hence
obtain the expression for the steady state probabilities and the average number of customers
in the system (AN)(May/June2014)
33. Four counters are being run on the frontiers of the country to check the passports of the
tourists. The tourists choose a counter at random. If the arrival at the frontier is poisson at
the rate  and the service time is exponential with parameter  /2, find the average queue
length at each counter. (AP)(May/ June 2014)
34. Customers arrive at the express checkout lane in a supermarket in a poisson process with a
rate of 15 per hour. The time to check out a customer is an exponential random variable with
mean of 2 minutes. Find the average number of customers present. What is the expected
waiting time for a customer in the system? (AP)(May/ June 2014)
35. The local one person barber shop can accommodate a maximum of 5 people at a time 4
waiting and 1 getting hair cut). Customers arrive according to a Poisson distribution with
mean 5 per hour. The barber cuts hair at an average rate of 4/hr (exponential service time) (i)
What percentage of time is the barber idle?(ii)What fraction of the potential customers are
turned away? (iii) What is the expected number of cutomers waiting for a hair-cut? (iv) how
much time can a customer expect to spend in the barber shop? (AP)(Nov./Dec.2014)
36. A tax consulting firm has 3 counter in its office to receive people who have problems
concerning their income, wealth and sales taxes. On the averages 48 persons arrive in an 8 hr
day. Each tax advisor spends 15 min. on the average upon arrival. IF the arrivals are poisson
distributed and service times are according to exponential distribution, find (i) the average
number of customers in the system (ii) the average number of customers waiting to be
serviced (iii) the average time a customer spends in the system?(AP)(April/ May 2015)
37. A small mail-order business has one telephone line and a facility for call waiting for two
additional customers. Orders arrive at the rate of one per minute and each order requires 2
min. and 30 seconds to take down the particulars. What is the expected number of calls
waiting in the queue? What is the mean waiting time in the queue? (AP) (April/ May 2015)
38. An airport has a single runway.. Airplanes have been found to arrive at the rate of 15 per
hour. It is estimated that each landing takes 3 minutes. Assuming a poisson process for
arrivals and an exponential distribution for landing time. Find the expected number of
airplanes waiting to land, expected waiting time. What is the probability that the waiting will
be more than 5 minutes? (AP)(April/ May 2015)
39. Customers arrive at a watch repair shop according to a poisson process at a rate of 1 per every
10 minutes, and the service time is an exponential random variable with mean 8 minutes.
Compute (1) The mean number of customers Ls in the system (2) The mean waiting time Ws
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of a customer spends in the system, (3) the mean waiting W q of a customer spends in the
queue and (4) the probability that the server is idle (AP)(May/ June 2016)
40. A petrol pump has 4 pumps. The service times follow the exponential distribution with mean
of 6 minutes and cars arrive for service is a Poisson process at the rate of 30 cars per hour.
Find the probability that no car is in the system? Find the probability that a customer has to
wait for service. Find the mean waiting time in the system? (AP)(May/ June 2016)
41. A one person barber shop has 6 chairs to accommodate people waiting for a haircut,. Assume
that customers who arrive when all the 6 chairs are full leave without entering the barber
shop. Customers arrive at the rate of 3 per hour and spend an average of 15 minutes in the
barber’s chair. Compute P0, Lq, P7 and Ws (AP)(May/ June 2016)
42. Consider a single server queue where the arrivals are poisson with rate   10 / hour . The
service distribution is exponential with   5 / hour . Suppose that customers balk at joining
the queue when it is too long. Specifically, when there are n in the system, an arriving
1
customer joins the queue with probability . Determine the steady state probability that
n 1
there are ‘n’ customers in the system. (AP)(May/June 2016)
43. The engineers have two terminals available to aid their calculations. The average computing
job requires 20 minutes of terminal time and each engineer requires some computations one
in half an hour. Assume that these are distributed according to an exponential distribution. If
the terminals can accommodate only 6 engineers in the waiting space find the expected
number of engineers in the computing center. (AP)(Nov./Dec.2015)
44. Find the system size probabilities for an M / M / C : FIFO /  /  queueing system under
steady state conditions. Also obtain the expression for average number of customers in the
system. (Nov./Dec.2015)(AN)
In a production shop of a company, the breakdown of the machines is found to be Poisson
with average rate of 3 machines per hour. Breakdown time at one machine costs Rs. 40 per
hour to the company. There are two choices before the company for hiring the repairman.
One of the repairmen is slow but cheap, the other fast but expensive. The slow repairman
demands Rs. 20 per hour and will repair the broken down machines exponentially at the rate
of 4 per hour. The fast repairman demands Rs. 30 per hour and will repair the machines
exponentially at an average rate of 6 per hour. Which repairman should the company hire?

COURSE OUTCOME: Acquire knowledge to characterize a model based on the Markovian

queuing models in different situation
UNIT V

NON-MARKOVIAN QUEUES

SYLLABUS: Finite source models - M/G/1 queue – PollaczekKhinchin formula - M/D/1 and
M/EK/1 as special cases – Series queues – Open Jackson networks.

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COURSE OBJECTIVE: Analyze probability and stochastic models which evolve with respect to
time in a probabilistic manner .Have an ability to design a model or a process to meet desired needs
within realistic constraints such as environmental conditions

PART-A

1. Explain few non-Markovian queues.

Solution:
Some of the queueing systems don’t have Markov property either in arrival or service pattern
and known as non-Markovian queues.

Some of the queues are M/Ek/1 – having erlang service time distributions M/D/1-constant
service time distributions.

2. State the P-K transform equation. (May/ June 2013)

Solution:

1   z  1LB  1  z 
Gz  
z  LB  1  z 

3. Write the P-K mean value formula.

Solution:

E N    

 2 1  C B2 
21   

4. Write the formula for number of customers in the non-Markovian queue with constant service
time.
2
LS   
21   

5. State the assumptions to derive P-K formula for non-Markovian queues.

Solution:
a) the server is not idle whenever a job is waiting for service.
b) the scheduling discipline does not base job sequencing on any priori information on job
execution times
c) the scheduling is no preemptive that is once a job is scheduled for service it is allowed to
complete without interruption.

6. Define open and closed queuing network.(May/ June 2013)(May/ June 2014)
Solution:
An open queuing network is characterized by one or more sources of job arrivals and
correspondingly one or more sinks that absorb jobs departing from the network. In a closed
network jobs neither enter nor depart from the network.

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7. Write the balance equation for a two-stage tandem network.

Solution:
 0  1    p1 pk 0 , k1    0 pk 0  1, k1  1  1 pk 0 , k1  1  pk 0  1, k1 
8. Draw a two-stage tandem network.

0 1

0 1

9. Write the solution for balance equation for a two-stage tandem network.
Solution: pk0 , k1   1   0  0k0 1  1 1k1
10. Define bottleneck of a system.
Solution:
As the arrival rate increases in a network of queues the node with larger with traffic intensity
will become instability and hence the node with largest traffic intensity is called ‘bottleneck’
of the system.

11. A two stage tandem network is such that the average service time of node 1 is 1 hour and the
average service time for node 2 is 2 hour and the arrival rate is 0.5 per hour. Find the
bottleneck of the system.
Solution:   0.5,  1,   2,   0.5,   0.25
1 2 1 2
Node 1 is bottleneck of the system.

12. A two stage tandem network is such that the average service time of node 1 is 1 hour and the
average service time for node 2 is 2 hour and the arrival rate is 0.5 per hour.
Solution:

  0.5,  1,   2,   0.5,   0.25

1 2 1 2
p  0,0   1  1   
   3
 1  2 8

13. Write the properties of Jackson network.

a. Arrivals from the “outside” to node I follow a Poisson process with mean rate i.
b. Service times at each channel at node I are independent and exponentially distributed with
parameter i.
c. The probability that a customer who has completed service at node i will go to next node j
is rij and rij indicates the probability that a customer will leave the system from node i.

14. State Jackson theorem for an open network. (Nov./ Dec. 2012)(Nov/Dec.2014)(April/ May
2015)
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In an open Jackson network, customers arrive at each station (node) both from outside the
system and from other stations. The customers may visit the various stations in any order and
may skip some stations. Each station has an infinite queue capacity and may have multiple
servers. A customer leaving station i goes to station j with probability p ij . So, the
probability that customer leaves the system after service at station ‘i’ is given
k
pio  1   pij
j 1

Let  j deonote the total arrival rate of customers to the station  j . Then  j can be got as
m j
the solution of  j  a j   i pij , j  1,2,....k where 1
i 1 cj j

15. Define series queues. (Nov./Dec.2013)

A series queue is one in which customers may arrive from outside the system at any node and
may leave the system from any node.

16. Define open network. (Nov./Dec.2013)

A network of K service facilities (or) nodes is called an open network if it satisfies the
following characteristics:

(a) Arrivals from outside to node ‘i’ followa poisson process with mean rate ri and join the
queue at ‘i’ and wait for his turn turn for service

(b) Service times at the channels at node ‘i’ are distributed with parameter i

(c) Once a customer gets the service completed at node ‘i’ he joins the queue at node ‘j’
with probability pij

17. State P-K formula for the average number in the system in a M/G/1 queueing model and
1
hence derive the same when the service time is constant with mean (May/ June 2014)


2
LS   
21   

18. What do you mean by Ek in the M/Ek/1 queueing model? (April/ May 2015)


Ek denotes the Erlang distributed service time pattern with the mean service time of


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19. What do the letter in the symbolic representation M/G/1 of a queueing model represent The
arrivals are Markovian and the service pattern is generally distributed (UD, ED, GD, RD, ED)
with single server. (April/ May 2015)

20. Write the expression for the traffic equation of the open Jackson queueing network (May/
June 2016)

Let i denote the total arrival rate of customers to the station and it can be got as the solution
n
of  j  ai   i pij
j 1

21. An M/D/1 queue has an arrival rate of 10 customers per second and a service rate of 20
customers per second. Compute the mean number of customers in the system. (May/ June
2016)

M/D/1 queue is with constant service time where var(T )  0

1
2 1 4  3  0.75 =1.
Ls     
2(1   ) 2 2( 1 ) 4
2

22. Derive the Pollaczek - Kintchine formula for the average number in the system when the
service time is constant with mean 1 . (Nov./Dec.2015)


23. Distinguish between and open and closed queueing network. (Nov./Dec.2015)

24. Pollaczek - Kintchine formula. (Nov./Dec.2015)

25. What do you mean by bottle neck of a network? (Nov./Dec.2015)

PART-B

1. Derive the Pollaczek-Kninchine formula for M/G/1 queue.(Nov/Dec 2007, 2008, 2010)
(April/May 2010)(Nov/ Dec.2012)(Nov/Dec.2014) (April/ May 2015) (Nov./Dec.2015) (AN)
2. Discuss M/G/1 queueing model and derive Pollaczek-Kninchine formula(Nov/Dec 2011) (AN)
3. Derive the P-K formula for the (M/G/1): (GD /  /  ) queueing model and hence deduce that
2 1
with the constant service time the P-K formula reduces to Ls    where  
2(1   ) E (T )

and   . (April/May 2012)(Nov./Dec.2013)(May/ June 2016) (AN)

4. In a heavy machine shop, the overhead crane is 75% utilized. Time study observations gave the
average slinging time as 10.5 minutes with a standard deviation of 8.8 minutes. What is the
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average calling rate for the services of the crane and what is the average delay in getting
service? If the average service time is cut to 8.0 minutes with a standard deviation of 6.0
minutes, how much reduction will occur on average in the delay of getting served? (AP)
5. A one-man barber shop takes exactly 25 minutes to complete one hair-cut. If customers arrive
at the barber shop in a Poisson fashion at an average rate of one every 40 minutes, how long on
the average a customer in the spends in the shop. Also find the average time a customer must
wait for service? (AP)(Nov./Dec.2013) (Nov./Dec.2015)
6. A patient who goes to a single doctor clinic for a general check up has to go through 4 phases.
The doctor takes on the average 4 minutes for each phase of the check up and the time taken for
each phase is exponentially distributed. If the arrivals of the patients at the clinic are
approximately Poisson at the average rate of 3 per hour, what is the average time spent by a
patient (i) in the examination (ii) waiting in the clinic? (AP)
7. A car wash facility operates with only one bay. Cars arrive according to a Poisson fashion with
a mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. The
parking lot is large enough to accommodate any number of cars. Find the average time a car
spends in the facility, if the time for washing and cleaning a car(1) is constant of 10 minutes (2)
is uniformly distributed between 8 and 12 minutes. (AP)(May/ June 2014)
8. A repair facility by a large number of machines has two sequential stations with respective rates
one per hour and two per hour. The cumulative failure rate of all the machines is 0.5 per hour.
Assuming that the system behavior may be approximated by the two-stage tandem queue,
determine the average repair time; determine the average number of customers in both stations
and the probability that both service stations are idle. (AP) (May/ June 2016) (Nov./Dec.2015)
9. An average of 120 minutes arrive each hour (inter-arrival times are exponential) at the
controller office to get their hall tickets. To complete the process, a candidate must pass
through three counters. Each counter consists of a single server, service times at each counter 1,
20 seconds; counter 2, 15 seconds and counter 3, 12 seconds. On the average how many
students will be present in the controller’s office. (AP) (April/May 2012)(May/ June 2014)
10. A car wash facility operates with only one bay. Cars arrive according to a Poisson fashion with
a mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. The
parking lot is large enough to accommodate any number of cars. Find the average number of
cars waiting in the parking lot, if the time for washing and cleaning a car follows a discrete
distribution with values equal to 4,8,15 minutes and corresponding probabilities 0.2, 0.6 and
0.2. (AP)
11. For a open queueing network with three nodes 1, 2 and 3, let customers arrive from outside the
system to node j according to a Poisson input process with parameters r j and let Pij denote the
proportion of customers departing from facility i to facility j. Given (r1 , r2 , r3 )  (1, 4, 3) and
 0 0 .6 0 .3 
 
Pij   0.1 0 0.3  determine the average arrival rate  j to the node j for j= 1, 2, 3.
 0 .4 0 .4 0 
 
V (AP)(Apr/May 2012)

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12. Automatic car wash facility operates with only one bay. Cars arrive according to a Poisson
distribution with a mean of 4 cars per hour and may wait in the facility’s parking lot if the bay
is busy. The parking lot is large enough to accommodate any number of cars. If the service time
for all cars is constant and equal to 10 minutes, determine
(1) mean number of customers in the system Ls
(2) mean number of customers in the queue Lq
(3) mean waiting time of a customer in the system Ws
(4) mean waiting time of a customer in the queue W q. (AP)(April/May2012)(May/June
2013)
13. In a bookshop, there are two sections, one for text books and the other for note books.
Customers from outside arrive at the text book section at a Poisson rate of 4 per hour and at
the note book section at a Poisson rate of 3 per hour. The service rates of the text book and
note book sections are respectively 8 and 10 per hour. A customer upon completion of service
at text book section is equally likely to go to the note book section or to leave the bookshop,
whereas a customer upon completion of service at note book section will go to the text book
section with probability 1 and will leave the bookshop otherwise. Find the joint steady state
3
probability that there are 4 customers in the text book section and 2 customers in the note
book section. Find also the average number of customers in the bookshop and the average
waiting time of a customer in the shop. Assume that there is only one salesman in each
section. (AP)
14. In an ophthalmic clinic, there are two sections- one section for assessing the power
approximately and the other for final assessment and prescription of glasses. Patients arrive at
the clinic in a Poisson fashion at the rate of 3 per hour. The assistant in the first section takes
nearly 15 minutes per patient and the doctor in the second section takes nearly 6 minutes per
patient. If the service times in the two sections are approximately exponential, find the
probability that there are 3 patients in the first section and 2 patients in the second section.
15. Derive the expected steady state system size for the single server queues with poisson input
and General service. (AN)(April/May 2011)
16. Write short notes on: (i) Series Queues (ii) Open and Closed Queue Networks (April/May
2011) (April/ May 2015) (U)
17. Explain how queueing theory could be used to study computer networks.(U)(April/May
2010)
18. Discuss open and closed networks. (U)(Nov/Dec 2011)
19. Write short notes on the following: (U) (Nov/Dec 2010)(Nov./Dec.2014)
(i) Queue networks
(ii) Series queues
(iii) Open networks
(iv) Closed networks
20. Consider a system of two servers where customers from outside the system arrive at server 1 at
a poisson rate 4 and at server 2 at a poisson rate 5. The service rates for server 1 and 2 are 8 and
10 respectively. A customer upon completion of service at server 1 is likely to go to server 2 or
leave the system; whereas a departure from server 2 will go to 25 % of the time to server 1 and
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will depart the system otherwise / determine the limiting probabilities Ls and Ws?(May/ June
2013) (Nov./Dec.2015) (AP)
21. Consider a two stage random queue with external arrival rate  to node ‘0’. Let  0 and 1 be
the service rates of the exponential servers at node ‘0’ and ‘1’ respectively. Arrival process is
poisson model this system using a Markov chain and obtains the balance equations.

22. Consider two servers. An average of 8 customers per hour arrive from outside at server 1 and
an average of 17 customers per hour arrive from outside at a server 2. Inter arrival times are
exponential. Server 1 can serve at an exponential rate of 20 customers per hour and server 2 can
serve at an exponential rate of 30 customers per hour. After completing service at station 1, half
the customers leave the system and half go to server 2. After completing service at station 2, ¾
of the customer complete service and ¼ return to server 1. Find the expected number of
customers at each server. Find the average time a customer spends in the system.
(Nov. /Dec.2012) (AP)

23. A car wash facility operates with only one bay. Cars arrive according to a poisson distribution
with mean of 4cars per hour and may wait in the facility’s parking lot if the bay is busy. The
parking lot is large enough to accommodate any number of cars. If the service time for a car has
uniform distribution between 8 and 12 minutes. Find (a) The average number of cars waiting in
the parking lot and (b) the average waiting time of a car in the parking lot. (Nov./Dec.2013)
(AP)
24. There are two salesmen in a ration shop one incharge of billing and receiving payment and the
other incharge of weighing and delivering the items. Due to limited availability of space, only
one customer is allowed to enter the shop, that too when the billing clerk is free. The customer
who has finished his billing job has to wait there until the delivery section becomes free. If the
customers arrive in accordance with a poisson process at rate 1 and the service times of two
clerks are independent and have exponential rate of 3 and 2 find (1) the proportion of customers
who enter the ration shop (2) the average number of customers in the shop (3) the average
amount of time that an entering customer spends in the shop. (Nov./Dec.2013) (AP)
25. Consider a open queueing network with parameter values shown below: (AP)

Facility j Sj j aj i 1 i2 i3

j 1 1 10 1 0 0.1 0.4

j2 2 10 4 0.6 0 0.4

j 3 1 10 3 0.3 0.3 0

(i) Find the steady state distribution of the number of customers at facility 1, facility2,and
facility3.
(ii) Find the expected total number of customers in the system.
(iii)Find the expected total waiting time for a customer May/June2014)(Nov./Dec.2015)
26. A repair facility shared by a large number of machines has 2 series stations with respective
service rates of 2 per hour and 3 per hour. If the average rate of arrivals is 1 per hour, find (1)

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the average number of machines in the system (2) the average waiting time in the system (3)
probability that both service stations are idle (April/ May 2015)(May/ June 2016) (AP)

27. Patients arrive at a clinic in a poisson fashion at the rate of 3 per hour. Each arriving patients
has to pass through two sections. The assistant in the first section take 15 minutes per patient
and the doctor in the second section takes nearly 6 minutes per patient. IF the service time in
two sections is approximately exponential find the probability that there are 3 patients in the
first sections and 2 patients in the second section. Find the average number of patients in the
clinic and the average waiting time of a patient? (AP)(April/ May 2015)
28. The police department has 5 petrol cars. A petrol car breaks down and repairs service one every
30 days. The police department has two repair workers, each of whom takes an average of 3
says to repair a car. Breakdown times and repair time are exponential. Determine the average
number of patrol cars in good condition; also find the average down time for a patrol car that
needs repairs. (AP)(May/ June 2016)
A Laundromat has 5 washing machines. A typical machine breaks down once every 5 days. A repair
takes an average of 2.5 days to repair a machine. Currently, there are three repair workers on duty.
The owner has the option of replacing them with a super worker, who can repair a machine in an
average of (5/6) day. The salary of the super worker equals the pay of the three repair workers.
Breakdown time and repair time are exponential. Should the Laundromat replace the three repairers
with a super worker? (Nov./Dec.2015) (AP)

COURSE OUTCOME: Able to analyze simple queuing networks, Model communication networks
and I/O computer systems

COURSE OUTCOMES

Course Name : PROBABILITY AND QUEUEING THEORY (MA6453)

Year/Semester : II / IV

Year of Study : 2016 –2017 (R – 2013)

On Completion of this course student will be able to

Understand the characteristics of probability distributions by identifying the
C210.1
discrete and continuous random variables.
Identify and inculcate adequate knowledge in multi random variables to
C210.2
understand covariance, correlation and transformation of random variables.
Analyze the relation between random input and output signals using the basics of
C210.3 random process and its characteristics to solve problems and model situations
using techniques of Markov process
Analyze basic properties of Markov chains and their applications in modeling
C210.4
queuing systems and acquire skills in analyzing Markovian queueing models.
Analyze probability and stochastic models which evolve with respect to time in a
C210.5 probabilistic manner and ability to analyze simple queuing networks , Model
communication networks and I/O computer systems

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CO – PO MATRIX:

CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
CO1 3 3 - - - - - - - - - 2
CO2
3 3 - - - - - - - - - 2

CO3
3 3 - - - - - - - - - 2

CO4
3 3 - - - - - - - - - 2

CO5
3 3 - - - - - - - - - 2

AVG
3 3 - - - - - - - - - 2

COURSE OUTCOMES AND PROGRAM SPECIFIC OUTCOMES MAPPING

CO – PSO MATRIX:

CO PSO1 PSO2 PSO3

CO1 - - --
CO2 - - -
CO3 - - -
CO4 - - -
CO5 - - -
AVG - - --

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