APPLIED PROBABILITY AND STATISTICS

UNIT I:RANDOM VARIABLES

1. A discrete random variable X has the probability function given below

X 0 1 2 3 4 5 6 7
P(X) 0 a 2a 2a 3a a2 2a 2 7a 2 +a
Find
(i) The value of a
(ii) P(X<6), P(X≥6)and P(0<X<4)
(iii) P(1.5<X<4.5/X>2)
(iv) The smallest value of λ for which P(X≤ λ)>½
(v) P(X<2),P(X>3),P(1<X<5)

1. A discrete random variable X has the probability function given below

X 0 1 2 3 4 5 6 7
2 2
P(X) 0 a 2a 2a 3a a 2a 7a 2 +a
Find the value of a.
2. A discr ete random variable X has the probability function given below
X 0 1 2 3 4 5 6 7
2 2
P(X) 0 a 2a 2a 3a a 2a 7a 2 +a
If a=1/10 then find P ( X< 2 ),P ( X > 3 ).
3. If the range of X is the set {0,1,2,3,4} and P(X=x)=0.2,determine the
mean and variance of the random variable
4. Let the random variable X denotes the sum obtained ‘m’ rolling a pair of fair dice.
Determine the probability mass function of X.
5. A discrete random variable X has the probability function given below
X 0 1 2 3 4 5 6 7
2 2
P(X) 0 a 2a 2a 3a a 2a 7a 2 +a
Find P(1.5<X<4.5/X>2) if a=1/10.

2. A random variable has the following probability distribution

X -2 -1 0 1 2 3
P(X) 0.1 K 0.2 2k 0.3 3k
Find
(i) the value of k
(ii) Evaluate P(X<2),P(-2<X<2),P(-2<X<3)
(iii) Find the cumulative distribution of X and
(iv) Evaluate the mean of X
(v) Evaluate the variance of X
(vi) P(X<3/X>1)

1. A random variable has the following probability distribution

X -2 -1 0 1 2 3
P(X) 0.1 k 0.2 2k 0.3 3k
Find the value of k.
2. A random variable has the following probability distribution
X -2 -1 0 1 2 3
P(X) 0.1 k 0.2 2k 0.3 3k
 Find P ( X< 3 / X > 1) if k = 1/15.
3. Define a discrete and continuous random variable.
4. Define probability density function.
5. A random variable has the following probability distribution
X -2 -1 0 1 2 3
P(X) 0.1 k 0.2 2k 0.3 3k
Evaluate the mean of X if k = 1/15.

3. (a) If the density function of a continuous R.V.X is given by

 ax , 0  x 1
 a ,1 x  2

f ( x)  
 3a  ax , 2  x  3
0 , elsewhere
(i) Find the value of a
(ii) Find the CDF of X
(iii) If x 1 ,x 2 ,x 3 are three independent observations of X, what is the
probability that exactly one of these 3 is greater than 1.5

(b) 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

1. If the density function of a continuous R.V.X is given by

 ax , 0  x 1
 a ,1 x  2

f ( x)  
 3a  ax , 2  x  3
0 , elsewhere
Find the value of a.
2. If a random variable X has the distribution function
1  e x for x  0
F ( x)  
0 for x  0
Where α is the parameter, then find P(1≤x≤2).
3. Is the cumulative distribution function F(x) of a random variable X always continuous?
Justify your answer.
4. If X has mean 4 and variance 9, Y has mean -2 and variance 5 and they
are independent. Find the variance of 2X+Y-5.
5. 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.
 2x , 0  x  1
4. (a) A random variable X has the pdf, f ( x)  
 0 , otherwise
(i) Find P(X<½),
(ii) P(¼<X<½),
(iii) P(X>3⁄4/X>½)
(b) Out of 800 families with 4 children each, how many families would be
expected to have
 (i) two boys and 2 girls,
(ii) atleast 1 boy
(iii) atmost 2 girls
(iv) Children of both sexes.
Assume equal probabilities for boys and girls.
 2x , 0  x  1
1. A random variable X has the pdf f ( x )  
 0 , otherwise
Find P(X< ½).
 k (1  x 2 ) , 0  x  1
2. Find k for the pdf of f ( x )  
 0 , otherwise
3. Out of 800 families with 4 children each, how many families would be
expected to have two boys and 2 girls.
4. Write any two properties of CDF F(x).
 2x , 0  x  1
5. A random variable X has the pdf, f ( x )  
 0 , otherwise
Find P(¼<X<½).
c e 2 x , 0  x  
5. (a) If the density function is of X equals f ( x )  
 0 , x0
Find c, what is P(X>2)?

(b) 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.
c e 2 x , 0  x  
1. If the density function is of x equals f ( x )  
 0 , x0
Find c.
 100
 , x  100
2. Check whether for f ( x )   x 2
 0 , x  100
is a probability density function.
3. The distribution function of a random variable X is given by
F(x) =1-(1+x)e - x : x ≥ 0, Find the mean.
4. If a random variable X takes the values 1,2,3,4, such that
2P(X=1)=3P(X=2)=P(X=3) =5P(X=4). Find the probability distribution
of X.
c e 2 x , 0  x  
5. If the density function is of X equals f ( x )  
 0 , x0
Find P(X>2)if c=2.
6. (a) By calculating the moment generating function of Poisson
distribution with parameter λ, prove that the mean and variance of
the Poisson distribution are equal .
 x , 0  x 1

(b) For the triangular distribution f ( x)   2  x , 1  x  2
0 , otherwise

 Find the mean, variance and the moment generating function.

1. Define Poisson distribution and State any two instances where Poisson
distribution may be successfully employed.
 x , 0  x 1

2. For the triangular distribution f ( x)   2  x , 1  x  2
0 , otherwise

Find the moment generating function.
3. Find the mean and variance of the distribution whose moment generating
function is (0.4e t +0.6) 2
 x e x , x  0
4. A random variable X has density function given by f ( x )  
 0, x  0
Find the moment generating function.
5. Find the mean of the Poisson distribution.

 2x , 0  x  1
7. (a) Let f ( x )  
 0 , otherwise
Find probability density function of Y=4X+2.
1 , x 1

If X is a continuous random variable with F  x   k  x  1 ,1  x  3
4
(b)
0 , x3

Find the value of k, probability density function and P(X <2).

1. If X is a continuous random variable with

1 , x 1

F  x   k  x  1 ,1  x  3
4

0 , x3

Find the value of k.
2. X and Y are independent random variables with variance 2 and 3. Find
the variance of 3X+4Y.
3. Define moment generating function.
4. Write down the properties of moment generating function.
5. If X is a continuous random variable with
1, x  1

F  x   k  x  1 ,1  x  3
4

0, x  3

Find P(X<2) if the value of k= 1/16.

8. (a) If the cumulative distribution functions of a R.V X is given by

ì 4
ï 1- 2 ; x > 2
F ( x) = í x
ï
î0 ;x £ 2
 Find (i)P(X<3), (ii) P(4<X<5), (iii) P(X ≥ 3)
 2x , 0  x  1
(b) If X is any continuous R.V having the pdf f ( x )  
 0 , otherwise
-x
and Y=e find the pdf of R.V Y.

1. If the cumulative distribution functions of a R.V X is given by

ì 4
ï 1- 2 ; x > 2
F ( x) = í x
ï
î0 ;x £ 2
Find P(x<3).
2. The moment generating function of the random variable X is given by
.
M x(t) = Find P(X=1).
3. The random variable X, has the pdf f(x) = e - x , 0<x<∞. Find the density
function of the variable Y=3X+5.
4. Find the distribution function of the random variable Y=g(x), in terms of
the distribution function of X, If it is given that
x-c for x>c
g(x)= 0 for |x|≤c
x+c for x<-c.
5. If the cumulative distribution functions of a R.V X is given by
ì 4
ï 1- 2 ; x > 2
F ( x) = í x
ï
î0 ;x £ 2
Find P(4<X<5).
ì 1 -x
ï e 2 ;x >0
9. (a) Let The random variable X has the pdf f ( x) = í 2
ï
ïî 0 ; otherwise
Find the moment generating function, mean, variance of X and also
find P(X>½).

(b) If X is a uniform random variable in the interval (-2,2). Find the pdf
of Y=X 2 .
ì 1 -x
ï e 2 ;x >0
1. Let the random variable X has the pdf f ( x) = í 2
ï
îï 0 ; otherwise
Find the mean.
2. If X is uniformly distributed over (-1,1) find the density function of

.
3. The first four moments of a distribution about X=4 are 1,4,10 and 45
respectively. Show that the mean is 5, variance is 3, μ 3 =0 and μ 4 =26.

4. If RV X has the mfg M x (t)= . Obtain the S.D. of X.

 ì 1 -x
ï e 2 ;x >0
5. Let The random variable X has the pdf f ( x) = í 2
ï
îï 0 ; otherwise
Find the moment generating function.

10. (a) Find the moment generating function of the random variable X
ì x -x
ï e 2 , x >0
having the pdf f ( x) = í 4 also deduct the first four
ï
ïî 0 , elsewhere
moment about the origin.

(b) Given the random variable X with density function

f(x)= 2x , 0 < x < 1
0 , elsewhere
Find the probability density function of Y=8X 3
1. Find the moment generating function of the random variable X having
ì x -x
ï e 2 , x >0
the p.d.f f ( x) = í 4
ï
ïî 0 , elsewhere
ì x -x
ï e2 , x >0
2. Let X be a random variable with pdf f ( x ) = í 4
ï
ïî 0 , elsewhere
Find E(X).
3. Define the r t h moment about the origin.
 2x , 0  x  1
4. Given the random variable X with density function f ( x )  
 0 , otherwise
Find the |dx/dy| if the pdf Y=8X 3 .
5. Find the first moment about the origin of the random variable X having
ì x -x
ï e 2 , x >0
the p.d.f f ( x) = í 4 .
ï
îï 0 , elsewhere
11. A continuous random variable X has the p.d.f given by f(x) = ce - | x | ,
   x   . Find the value of C, moment generating function of X and the
F(x).

1. A continuous random variable X has the p.d.f given by f(x) = ce - | x | .

   x   . Find the value of C.
   
2. If X is uniformly distributed in  ,  find the probability distribution
 2 2
function of y = tan x.
3. If the pdf of X is f X (x) =e - x , x>0 find the pdf of Y=2X+1.
4. A continuous random variable X has probability density function given
by f(x)=3x 2 , 0≤ x ≤ 1. Find k such that P(X>k) =0.05.
 5. A continuous random variable X has the pdf given by f(x) = ce - | x | .
   x   . Find the moment generating function of X if c=1/2.

12. (a) Find the moment generating function of the binomial random
variable with parameters m and p and hence find its mean and
variance.
ì -x
ï 2 , x³ 0
(b) If P( x ) = í xe
ï 0 , x <0
î
i. Show that P(x) is a pdf.
ii. Find F(x)

1. The sum of two independent Binomial variate is not a Binomial variate.

2. Comment the following: the mean of a binomial distribution is 3 and
variance is 4.
3. Define the Binomial Distribution.
4. For a binomial distribution mean is 6 and standard deviation is . Find
the first two terms of the distribution.
ì -x
ï 2 , x³ 0
5. If P( x ) = í xe
ï 0 , x <0
î
Show that p(x) is a pdf.

13. (a) The number of monthly breakdown of a computer is 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 and
(iii) with at least one breakdown.

(b) A continuous random variable X has P.D.F f(x)= kx 2 e - x ; X≥0, Find k,

r t h moment, mean and variance.
1. The number of monthly breakdown of a computer is random variable
having a Poisson distribution with mean equal to 1.8. Find the
probability that this computer will function for a month without a
breakdown.
2. On an average typist makes 2 mistakes per page. What is the probability
that she will make exactly 3 or more errors on a page?
3. If X is a Poisson vitiate such that P(X=2) =9P(X=4) +90PX=6) find the
variance.
4. A continuous random variable X has pdf f(x) = kx 2 e - x ; x ≥ 0, Find k and
r t h moment.
5. A continuous random variable x has pdf f(x) = kx 2 e - x ; x ≥ 0, Find mean
if k=1/2.

14. (a) 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?

(b) State and prove memory less property of geometric distribution. Also
derive mean and variance of a G.D.

1. One percent of jobs arriving at a computer system need to wait until

weekends for scheduling, owing to core–size limitations. Find the
probability that among a sample of 200 jobs there are no jobs that have
to wait until weekends.
2. The sum of two independent Poisson variate is a Poisson variate.
3. A machine manufacturing screws is known to produce 5% defective. In a
random sample of 15 screws, what is the probability that there are
exactly 3 defectives?
4. In which Probability distribution, variance and mean are equal. And also
write the variance.
5. A machine manufacturing screws is known to produce 5% defective. In a
random sample of 15 screws, what is the probability that there are not
more than 3 defectives?

15. (a) A die is tossed until 6 appears. What is the probability that it must
be tossed more than 4 times.

(b) 6 dice are thrown 729 times. How many times do you expect at least
three dice to show 5 or 6?

1. In a company, 5% defective components are produced. What is the

probability that at least 5 components are to be examined in order to get
3 defectives?
2. Obtain the mean for a Geometric random variable.
3. What is meant by memory less property? Which continuous distribution
follows this property?
4. Define Geometric distribution.
5. Find the variance of the Geometric distribution.

16. (a) 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 is
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?
(b) Find the mean, variance and moment generating function of a
random variable uniformly distributed in the interval (a, b).

1. If X is geometric variate taking values 1,2,3….∞.Find P(X is odd).

2. Find the mean of uniform distribution.
3. If the probability that an applicant for a driver’s license will pass the
road test on any given trial is 0.8. What is the probability that he will
finally pass the test on the fourth trial?
4. Find the moment generating function of uniform distribution.
5. Find the variance of the Uniform distribution.
17. (a) Buses arrive at a specified stop at 15 min. interval starting at 7 AM,
That is, they arrive at 7, 7.15, 7.30, 7.45 and so on. If a passenger
arrives at the stop at random time that is uniformly distributed
between 7 and 7.30 A.M. find the probability that he waits.
(i) less than 5 min for a bus and
(ii) at least 12 min for a bus.

(b) The mileage which car owners get with a 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 least20,000 km
(ii) at least 30,000 km.

1. Suppose that a bus arrives at a station every day between 10.00 am and
10.30 am at random. Let X is the arrival time. Find the distribution
function of X and sketch its graph.
2. If X is a random variable uniformly distributed in (0,1), find the pdf of
Y = sin X.
3. Define the Uniform Distribution function.
4. What is the pdf of an exponential distribution?
5. The mileage which car owners get with a 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 atleast 20,000
km.

18. (a) The mean yield for one-acre plot is 662 kilos with a S.D. 32 kilos,
Assuming normal distribution, how many one-acre plots in a batch of
1,000 plots would you expect to have yield
(i) over 700 kilos
(ii) below 650 kilos
(iii) what is the lowest yield of the best 100 plots?

(b) Find the M.G.F, Mean and Variance for Normal distribution.

1. Define Generalised form of the Gamma distribution?

2. The marks obtained by a number of students for a certain subject are
assumed to be approximately normally distributed with mean value 65
and with a S.D. of 5. What is the probability that the student will have
marks over 70?
3. Define the normal distribution.
4. A sample of 100 items is taken at random from a batch known to contain
40% defectives. What is the probability that the sample contains: (i)
atleast 44 defectives, (ii) exactly 44 defectives.
5. Define expectation of a random variable.

19. (a) The time (in Hours) required to repair a machine is exponentially
distributed with parameter λ=½. What is the probability that the
repair time exceeds 2 h? What is the conditional probability that a
repair takes at least 10 h given that its duration exceeds 9h?
 (b) The local authorities in a certain city install 10,000 electric lamps in
the streets of the city. If these lamps have an average life of 1000
burning hours with a S.D. of 200 hours, assuming normality, what
number of lamps might be expected to fail
(i) in the first 800 burning hours
(ii) between 800 and 1200 burning hours

1. The time (in Hours) required to repair a machine is exponentially

distributed with parameter λ=½. What is the conditional probability that
a repair takes at least 10 h given that its duration exceeds 9h?
2. The time (in Hours) required to repair a machine is exponentially
distributed with parameter λ=½. What is the probability that the repair
time exceeds 2 h?
3. Define Binomial frequency distribution.
4. If X has an exponential distribution with parameter λ, find the
probability density function of Y=logX.
5. Find the mean of the Exponential distribution.

20. (a) Find the m.g.f, mean and variance of exponential distribution.

(b) Prove that the Poisson distribution as the limiting case of binomial
distribution.

1. In which distribution as a limiting case of binomial distribution. And

also write that the parameter of the distribution.
2. If X is uniformly distributed random variable with mean 1 and variance
4/3. Find P(X<0).
3. If X is uniformly distributed over (0, 10), calculate P(X<3), P(X>6).
4. Prove the Memory less property of exponential distribution.
5. In a normal distribution 31% of the items are under 45 and 8% are over
64. Find the mean and the S.D.