1.a) State and prove Addition law of Probability.

b) A fair coin is tossed 3 times. Let X be a random variable that denotes the number of
heads appearing in each outcome. Find and plot the CDF of X.

2.a) Consider the experiment of tossing two dice simultaneously. If a random variable is
defined as X=sum of the two faces, find and plot the pdf of X.
b) A binary communication system transmits two messages X=+1 and X=-1, with equal
probability. At the receiver the messages received can be Y= +1 or 0 or -1. Let
P(Y= -1/X= +1)=0.1; P(Y=+1/X= -1)= 0.2; P(Y=0/X= +1)=P(Y= 0/X= -1)=0.05. Find
the probability P(X=0/Y=0).

3.a) Verify that Rayleigh Density is a valid density Function.

b) Find and plot the CDF of uniform random variable distributed over (a, b).

4.a) Find the density of the random variable Y=2X+3, where X is a uniform random
Variable over (-2, 3).
b) A fair coin is tossed 10 times. Find the probability of getting the chance of Head
6 times? [8+7]

5. Find the density of the random variable Z=X+Y, where X and Y are two independent
uniform random variables over (-1, 1). [15]

6. X and Y are two random variables defined as X=Cosφ and Y=Sinφ where ‘φ’ is a
uniform random variable over (0,2π). a) Verify that X and Y are uncorrelated
b) Check X and Y for independence. [7+8]

7.a) X(t)=ACos(wt) is a random process, where ‘A’ is uniform random variable over (0, π).
Check X(t) for stationary.

b) X(t)=2.Cos(2πt+Y) is a ‘Y’ is a discrete random variable taking values ‘0’, and ‘π/2’
with equal probability. Find the Mean of X(t) and RXX(0,1). [7+8]

8. X(t) = A. Cos(wt +φ) is a random process where ‘φ’ is uniform random variable over
(-π,π) and ‘A’ is a normal random variable with zero mean and unity variance and is
independent of ‘φ’. Find the Autocorrelation function of X(t). [15]
Code No: 153BQ
R18
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
B.Tech II Year I Semester Examinations, October - 2020
PROBABILITY THEORY AND STOCHASTIC PROCESSES
(Electronics and Communication Engineering)
Time: 2 hours Max. Marks: 75
Answer any five questions
All questions carry equal
marks
---

1.a) Write about the Binomial and Poisson distributions with their characteristics.
b) If A and B are independent events, prove that the events A′ and B, A and B, and A and B′
are also independent. [7+8]

2.a) If X and Y are two random variables which are Gaussian. If a random variable Z
is defined as Z=X+Y, Find fZ(Z).
b) State and prove Bayes theorem of probability. [7+8]

3.a) Distinguish between the monotonic and non-monotonic transformations.

b) Prove that the variance of a weighted sum of uncorrected random variables equals the
weighted sum of the variances of the random variables. [6+9]

4.a) Find the moment generating function about origin of the Poisson distribution.
b) Define conditional distribution and density function of two random variables X and Y.
[8+7]

5.a) Discuss about the autocorrelation function and its properties

b) Let X be a random variable defined, Find E [3X] and E[X2] given the density function

as fx
 x  16cos x 8 4  x  4
[7+8]
  0 elsewhere



6. A Gaussian random variable X having a mean value of zero and variance one is
transformed to another random variable Y by a square law transformation. Find the
density function of Y. [15]

7.a) Discuss the relation between PSDs of input and output random process of an LTI system.
b) Evaluate the PSD of a random process z(t) = X(t)+y(t) where x(t) and y(t) are zero mean,
individual random process. [8+7]

8.a) Discuss about the Entropy and Information rate and their measurement parameters.
b) Explain the Source coding and mention the process of Huffman coding with an example.
[7+8]
1.a) A random variable X has probability density function
5𝑒−5 ; 0 ≤ 𝑥 ≤ ∞
𝑓𝑥 (𝑥) = {
0 ; 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒
(
Find 𝑖) 𝑋 ) 2
𝑖𝑖)𝐸[(𝑋 − 1) ].
b) Derive expressions for mean and variance for Gaussian variable. [8+7]

2.a) State and prove any four properties of distribution function?

b) A missile can be accidentally launched if two relays A and B both have failed. The
probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also
known that B is more likely to fail (probability 0.06) if A failed.
i) What is the probability of an accidental missile launch?
ii) What is the probability that A will fail if B has failed?
iii) Are events “A fails” and “B fails” statistically independent? [8+7]

3.a) Show that the process 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔0𝑡 + 𝜃) is wide sense stationery if it is assumed
that 𝐴 𝑎𝑛𝑑 𝜔0 are constants and 𝜃 is random variable which is uniformly distributed over
interval [0,2𝜋].
b) Write the conditions for a Wide sense stationary random process. [8+7]

4.a) Explain how random processes are classified with neat sketches.
b) The power Spectral density of (𝑡)is given by 𝑆𝑥𝑥 (𝑤) = 1/1 + 𝑤2 for 𝑤 > 0. Find the
autocorrelation function. [7+8]

5.a) Explain the properties of joint distribution

b) Let 𝑓𝑥𝑦 (𝑥, 𝑦) = 𝑥 = 𝑦 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1 = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. Find the conditional
density of the following: i) X given Y (ii) Y given X. [7+8]

6.a) Find the probability of the event {𝑋 ≤ 5.5} for Gaussian random variable having 𝑎𝑋 = 3
and 𝜎𝑥 = 2.
b) Find auto correlation function of a random process whose power spectral density is given
4
by 𝜔 2 [8+7]
1+
4

7.a) For two jointly stationary random processes, the cross-correlation function is
𝑅𝑋𝑌 (𝑟) = 2𝑒−2𝑟 (𝑟). Find the two cross-spectral density function.
b) The power spectral density of a stationary random process is given by

SXX  k    k
  
A; otherwise


Find the autocorrelation function. [8+7]
8. Show that a narrow band noise process can be expressed as in-phase and quadrature
components of it. [15]
 PART - A
(25 Marks)
1.a) Write the conditions for a function to be a random variable. [2]
b) Explain the significance of mathematical model of experiments. [3]
c) Write short notes on Chebychev’s inequality. [2]
d) Define Characteristic function and present generation of moments using it. [3]
e) State central limit theorem for the case of equal distributions. [2]
f) Write the properties of jointly Gaussian random variables. [3]
g) What is a WSS random process? [2]
h) Write short notes on Gaussian random process. [3]
i) Write the expression for power spectral density. [2]
j) Write any three properties of cross-power density spectrum. [3]

PART - B
(50 Marks)
2. A missile can be accidentally launched if two relays A and B both have failed. The
probabilities of A and B failing are known to be 0.01 and 0.03, respectively. It is
also known that B is more likely to fail (probability 0.06), if A has failed.
a) What is the probability of an accidental missile launch?
b) What is the probability that A will fail, if B has failed?
c) Are the events “A fails” and “B fails” statistically independent? [10]
OR
3. You (A) and two others (B and C) each toss a fair coin in a two-step gambling
game. In step1 the person whose toss is not a match to either of other two is “odd
man out”. Only the remaining two whose coins match go on to step2 to resolve the
ultimate winner.
a) What is the probability that you will advance to step2 after the first toss?
b) What is the probability you will be out after the first toss?
c) What is the probability that no one will be out after the first toss? [10]

4.a) Obtain the moment generating function of a uniformly distributed random variable.
b) Obtain the variance of Raleigh random variable. [5+5]
OR
5.a) A random variable X uniformly distributed in the interval (0, π/2). Consider the
transformation Y=sinx, obtain the pdf of Y.
b) Obtain the variance of Gaussian random variable. [5+5]
6.a) The joint characteristic function of two random variables is given by
XY(ω1,ω2) = exp(-ω 12- 4ω 22) . Check whether X and Y are uncorrelated or not.
b) X and Y are statistically independent random variables and W = X+Y obtain the pdf
of W. [5+5]
OR
7.a) Write the properties of joint distribution function.
b) Prove that the variance of weighted sum of N random variables equals the weighted
sum of all their covariances. [5+5]

8. Define autocorrelation function of a random process. Write properties of auto

correlation function of a WSS process and prove any three of them. [10]
OR
9.a) A random process X(t) = Acos(ωot) + B sin(ωot) where ωo is a constant and A, B
are uncorrelated zero mean random variables with same variances. Check whether
X(t) is WSS or not?
b) Classify random processes and explain. [5+5]

10. Derive the relationship between cross-power spectrum and cross-correlation

function. [10]
OR
11.a) The autocorrelation function of a random process R XX(τ) = 4 cos(ωoτ), where ωo is a
constant. Obtain its power spectral density.
b) Obtain the average power in the random process X(t) = Acos(ω ot+) where A, ωo
are real constants and  is a random variable uniformly distributed in the range
(0, 2π). [5+5]
1.a) Define and explain the concepts of Joint and Conditional probability.
b) How do you explain statistically independent events using Baye’s rule?
c) A bag contains four balls. Two balls are drawn and are found to be white. Find the
probability that all the balls are white. [5+5+5]

2.a) Define conditional distribution and density function of two random variables X and Y.
b) State and prove any three properties of moment generating function. [7+8]

3.a) Define and explain conditional probability mass function. Give its properties.
b) The joint distribution of X and Y is given by
 x 2  y 2 
f  x   4xye , x  0, y  0
Show that X and Y are independent random variables. [7+8]

4.a) X(t) is a stationary random process with a mean of 3 and an auto correlation function of
6+5 exp(-0.2 |τ|). Find the second central Moment of the random variable Y = Z-W,
where Z and W are the samples of the random process at t = 4 sec and t = 8 sec
respectively.
b) Find the cross correlation between the processes X(t) = Acos(ωt) + Bsin(ωt) and
Y(t) = Bcos(ωt) - Asin(ωt), Where A and B are two standardized Gaussian random
variables. [8+7]

5. A stationery random process X(t) has spectral density SXX(ω) = 25/(ω2+25) and an
independent stationery process Y(t) has the spectral density S YY(ω) = ω2/(ω2+25). If X(t)
and Y(t) are zero mean, find the:
a) PSD of Z(t) = X(t) + Y(t)
b) Cross spectral density of X(t) and Z(t). State and prove the results used. [7+8]

6.a) Distinguish between ensemble average and time average of a random process.
b) A random process is defined as X(t) = A sin(ωt + θ), where A is a constant and θ is a
random variable uniformly distributed over (-π, π). Check X(t) for stationarity. [7+8]

7.a) For two jointly stationary random processes, the cross-correlation function is
RXY(τ) = 2e-2τ u(τ). Find the two cross-spectral density functions.
b) List the properties of cross power spectral density function. [7+8]

8.a) Show that the autocorrelation function of a stationary random process is an even function
of τ.
b) Discuss the properties of conditional distribution function. [7+8]

---ooOoo---
 PART- A
(25 Marks)
1.a) When two dice are thrown simultaneously, if X and Y denote the numbers on the first
and second respectively, find the probability for X+Y to be greater than or equal to 8.
[2]
b) A box contains three coins: one is fair, one is two headed and one coin is weighted so that
the probability of head is 1/3. A coin is selected at random and tossed. Find the
probability for head to appear. [3]
c) A random variable X is having a CDF as shown:

Find the density of X over( -2, 1) and P(X=1.2). [2]

 x for 0  x  1
d) The density function of a random variable X is f x 2  x for 1  x  2 . Find its
 
 for x  2 
 0
CDF. [3]
e) A fair coin is tossed three times. Let X denote a „0‟ or „1‟ according a head or tail occurs
in the first toss and let Y denote the number of heads which occur. Write the joint
probability Matrix of X and Y. [2]
f) The joint density of two random variables X and Y is given as

f x, y  0.21  x 1  y  2  0.35 x 1  y  5  0.14 x 1  y  8  0.09 x  2  y

 2
0.15  x  2  y  5  0.06  x  2  y  8
Find P(Y is even, given X is even). [3]
g) State and prove the Periodicity property of Auto correlation function of a periodic WSS
process. [2]
 h) If X(t) and Y(t) are two independent Stationary Processes, find the Auto correlation
function of the process Z(t)=2.X(t).Y(t). [3]
i) Verify that Cross Spectral Density of two Uncorrelated and Stationary Random processes
is an impulse function. [2]
j) The input to an LTI system with Impulse response h(t)=δ(t)+t2.exp(-at).U(t) is a WSS
process with mean of 3.Find the mean of the output of the system. [3]

PART-B
(50 Marks)
2.a) Three machines A, B and C produce 55%, 25% and 20% of the total number of items of a
factory. The percentage of defective output of these machines are 3%, 2% and 4%. If an
item is selected at random, (i) find the probability that the item is defective
(ii) If the selected item is defective, find the probability that the item is produced by
Machine B.
b) In a single throw of a pair of dice, what is the probability of obtaining the sum of two
faces of the dice is equal to at least 10. [6+4]
OR
3.a) Two different digits are selected at random from the digits 1 to 9. (i) If the sum of the
digits is odd, what is the probability that 2 is one of the digits selected? (ii) If 2 is one of
the selected digits, what is the probability for the sum to odd.
b) A binary communication system transmits a 0 and 1 with equal probabilities. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 1/8 and a
transmitted 0 is received as 1 with a probability of 3/4. (i) Find the probability for the
transmitted message to be a 1. (ii) If a one is received, find the probability that the
transmitted is a 1. [6+4]

4.a) A discrete random variable is having a density function given by

f(x)=0.1δ(x)+0.4δ(x-1)+0.3δ(x-2)+0.2 δ(x-3).
x3
Find the density of the random variable Y  2  x2  .
3
b) Verify that Gaussian Density function is a valid density function. [5+5]
OR
5.a) Find the Mean and Mean square value of Rayleigh Random Variable.
b) X is a Gaussian random variable with zero mean and unity variance. Find the density of
the random variable Y = X for x ≥ 0, and Y=0 for x < 0. [5+5]
5
6.a) Two random variables X and Y have a joint probability density function f(x,y)= 2
16 x y
for 0 < y < x < 2; and zero elsewhere. Check X and Y for independence.
b) Let f(x,y) = x + y for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; and zero elsewhere. Find the density of X,
given Y. [6+4]
OR
7.a) The joint probability mass function of two random variables X and Y is
P(x,y)=K(2x+3y) for x=0,1,2; and y=1,2,3. Find the marginal probability distributions of
X and Y. Find all the possible conditional probabilities of X.
b) If X is a random variable with mean 3 and variance 2, verify that the random variables
„X‟ and Y=-6X+22 are orthogonal. [6+4]
8.a) A Random Process X(t)=A.Cos (2π fc t) , where A is a Gaussian Random Variable with
zero mean and unity variance, is applied to an ideal integrator, that integrates with respect
to „t‟, over (0,t). Check the output of the integrator for stationarity.
b) In the random Process X(t)=A.CosWt, A is uniform random Variable over (0,1) and W is
a constant. Find the Auto correlation function of X(t). [6+4]
OR
9.a) X(t) is a random process with mean =3 and Autocorrelation function
Rxx(τ) =10.[exp(- 0.3|τ|)+2]. Find the second central Moment of the random variable
Y=X(3)-X(5).
b) X(t) is a WSS process and Y(t)=A.Cos(Wct+θ) is a random process which is independent
of X(t). Here, θ is a uniform random variable over(-π,π).If the Auto correlation function
of X(t) is Rxx(τ), find the Auto correlation of Z(t)=X(t).Y(t). [5+5]

10.a) Find the Auto correlation and PSD and M.S. Value of the random process, X(t)= m(t).
Cos(Wt+Ф), where m(t) is a WSS process and „Ф‟ is a uniform random variable over
(0,2π), and is independent of m(t).
b) A noise process with zero mean and of PSD “K” is applied to an R-L LPF. Find the Mean
Square value of the output Process. [5+5]
OR
11.a) Let x(t)=Y(t) -Y(t-2) is a random process, where Y(t) is also a stationary random process.
It is given Var(X(t))=20.Var(Y(t)). Find RYY(2)/var(X(t)) and also BXX.
b) X(t)=A.Sin(wt+θ) is a random process, with „θ‟ being a uniform random variable over
the interval (-π, π). If Y(t)=(1/2).X(t). Are X(t) and Y(t) are jointly stationary and find
Syy.
[5+5]

---oo0oo---
 PART- A
(25 Marks)
1.a) A box contains nine cards numbered through 1 to 9, and B contains five cards
numbered through 1 to 5. If a box is chosen at random, and a card is drawn which
even numbered, what is the probability for the card to be from box A. [2]
b) Let a die be weighted such that the probability of getting numbers from 2 to 6 is
that number of times of probability of getting a1. When the die thrown, what is
the probability of getting an even or prime number occurs. [3]
c) Find the CDF of a random variable X, uniform over (-3, 3). [2]
d) The density of a random variable X is given as f(x)= K[U(x)-U(x-4)]+0.25δ(x-2).
Find the probability of X ≤ 3. [3]
e) X and Y are discrete random variables and their joint occurrence is given as
X\Y 1 2 3
1 1/18 1/9 1/6
2 1/9 1/18 1/9
3 1/6 1/6 1/18
Find the Conditional Mean of X, given Y=2. [2]
f) X and Y are two uncorrelated random variables with same variance. If the random
variables U=X+ kY and V=X+(σx/σy)Y are uncorrelated, find K. [3]
g) State and prove the Periodicity Property of Auto Correlation function of a
Stationary Random Process. [2]
h) If X(t) is a Gaussian Random Process with a mean 2 and exp (-0.2|τ|). Find the
Probability of X(1) ≤ 1. [3]
i) Verify that the cross spectral density of two uncorrelated stationary random
processes is an impulse function. [2]
j) The output of a filter is given by Y(t)=X(t+T)+X(t-T), where X(t) is a WSS
process, power spectral density Sxx(w), and T is a constant. Find the power
spectrum of Y(t). [3]
PART-B
(50 Marks)

2.a) Consider the experiment of tossing two dice simultaneously. If X denotes the sum
of two faces, find the probability for X ≤ 6.
b) A fair coin is tossed 4 times. Find the probability for the longest string of heads
appearing to be three as a result of the above experiment.
c) In certain college, 25% of the boys and 10% of the girls are studying
Mathematics. The girls constitute 60% of the student body. If a student is
selected at random and studying mathematics, determine the probability that the
student is a girl. [3+3+4]
OR
3.a) Coin A has a probability of head =1/4 and coin B is a fair coin. Each coin is
flipped four times. If X is the number of heads resulting from coin and Y denotes
the same from coin B, what is the probability for X=Y?
b) A dice is thrown 6 times. Find the probability that a face 3 will occur at least two
times. [6+4]

4.a) Find the Moment generating function of a uniform random variable distribute
over (A, B) and find its first and second moments about origin, from the Moment
generating function.
b) A random variable X has a mean of 10 and variance of 9. Find the lower bound on
the probability of (5<X<15). [5+5]
OR
5.a) Find the Moment generating function of a random variable X with density
function
 x, for 0  x  1 
f  x  2  x, for 1  x  2
 
 
 0, else where 
b) If X is a Gaussian random variable N(m, σ2 ), find the density of Y=PX+Q, where
P and Q are constants. [5+5]

6.a) If X1,X2,X3, - - - - - Xn are ‘n’ number of independent and Identically distributed

random variables, such that Xk = 1 with a probability 1/2; = -1 with a probability
1/2. Find the Characteristic Function of the random Variable Y= X1+X2+X3+ - - -
+ Xn.
b) If Independent Random Variables X and Y both of zero mean, have variance 20
and 8 respectively, find the correlation coefficient between the random Variables
X+Y and X-Y. [5+5]
OR
7.a) Let X=Cosθ and Y=Sinθ , be two random variables, where θ is also a uniform
random variable over (0,2π). Show that X and Y are uncorrelated and not
independent.
b) If X is a random variable with mean 3 and variance 2, verify that the random
Variables ‘X’ and Y= -6X+22 are orthogonal. [6+4]

8.a) X(t) is a random process with mean =3 and Autocorrelation function

Rxx(τ) =10.[exp(- 0.3|τ|)+2]. Find the second central Moment of the random
variable Y=X(3)-X(5).
b) X(t)=2ACos(Wct+2θ) is a random Process, where ‘θ’ is a uniform random
variable, over (0,2π). Check the process for mean ergodicity. [5+5]
OR
9.a) A Random Process X(t)=A.Cos (2π fc t) , where A is a Gaussian Random
Variable with zero mean and unity variance, is applied to an ideal integrator, that
integrates with respect to ‘t’, over (0,t). Check the output of the integrator for
stationarity.
b) A random Process is defined as X(t)=3.Cos(2πt+Y), where Y is a random
Variable with p(Y=0)=p(Y=π)=1/2.Find the mean and Variance of the Random
Variable X(2). [5+5]
10.a) Find and plot the Autocorrelation function of (i) Wide band white noise
(ii) Band Pass White noise.
b) Derive the expression for the Cross Spectral Density of the input Process X(t)
and the output process Y(t) of an LTI system in terms of its Transfer function.
[5+5]
OR
11.a) Compare and contrast Auto and cross correlations.
b) If Y(t) = A.Cos(w0t+θ)+N(t), where ‘θ’ is a uniform random variable over (-π,π),
and N(t) is a band limited Gaussian white noise process with PSD=K/2. If ‘θ’ and
N(t) are independent, find the PSD of Y(t). [4+6]

---ooOoo---
 PART- A
(25 Marks)
1.a) The joint occurrence of two events A and B is given as
 A B 1 2 3 
1 1 12 1 6 1 12
P  A, B    . Find the probability of B is even, given that
 2 1 6 1 4 1 12
3 1 12 1 12 0
  
A is even. [2]
b) A die is weighted so that even numbers have same chance of appearing, and the
odd numbers have the same chance of appearing, and each even number is twice
as likely to appear as any odd number. When the die is tossed, what is the
probability of getting an even numbered face. [3]
c) A random variable X has a CDF given by
 0; for x  0

F  x   cos x . Find  x  2  , and justify the answer. [2]
X 1  0x
; for
FX

 2
d) A random variable X has the following distribution:

xi 0 1 2 3 4 5 6 7 8
P(xi) a 3a 5a 7a 9a 11a 13a 15a 17a

Find the smallest value of x, for which P[(X ≤ x) > 0.5]. [3]
e) A fair coin is tossed three times. Let X denote a ‘0’ or a ‘1’ according a head or
tail occurs in first toss and let Y denote the number of heads which occur. Find
the joint distribution of X and Y. [2]
f) Find the correlation coefficient between random variables X and Y related as
X=20Y. [3]
g) A random process is defined as X(t)= A.Coswt, where A is a uniform random
variable. What is the condition on A for the process to be WSS? [2]
h) X(t)=A Cos(2πt+Y) is a random process, where Y is a random variable such that
P(Y=0)=1/2 and P(Y=π/2)= 1/2. Find the correlation between the random
variables X(0) and X(1). [3]
i) A random process with PSD of K watts/Hz is applied to an ideal LPF with pass
band over (-B Hz to +B Hz). Find the noise power at the output of the filter. [2]
j) Find the cross spectral density of two uncorrelated stationary random processes
X(t) and Y(t). [3]
 PART-B
(50 Marks)
2.a) A coin is flipped three times and X denotes the number of heads that show up.
The probability of getting a head in each flip is ‘q’. Give the probability
distribution of X. Find the probability with which X>1.
b) A pair of fair dice is thrown. Find the probability that the sum is 10 or greater if
(i) 5 appears on the first dice (ii) 5 appears at least on one of the dice. [5+5]
OR
3.a) Find the probability of getting a total of 4 at least once in 3 tosses of a pair of
dice.
b) Two different digits are selected at random from digits 1 to 9. If the sum of the
digits selected is odd, what is the probability for 2 to be one of the numbers
selected?
c) A pair of fair dice is tossed. Find the probability that the maximum of the two
numbers is greater than four. [3+4+3]

4.a) The pdf of a random variable X is given by

f  x   K u  x   u  x  6  0.25  x  3  0.25  x  6 . Find and plot
the CDF of X.
b) The density function of a random variable X is given as
 ax; for 0  x  1 
 a; for 1  x  2 
f  x    . If three independent observations are made on
 3a  ax; for 2  x  3 
 0; else where 
X, find the probability with which exactly one of them is greater than 1.5. [5+5]
OR
5.a) Find the density of the Random variable Y=Tan(X), where X is uniform random
variable over ( -π/2, π/2).
b) For a random variable X with pdf f(x)= 0.5 exp(-|x|), find the probability that
an observed value will fall within ±2σ2 of the mean value. [4+6]

6.a) Find the density of Z=X+Y, where X and Y are two independent random
variables, which are uniform over (-2, +2) and (-1, 3) respectively.
b) A product is classified according to the number of defects it contains (X) and the
factory (Y) producing the product. The joint probability distribution is given by
 X Y 1 1 
0 1 8 1 16
 
 1 1 16 1 16
2 3 16 1 8
 
 3 1 8 1 4 
(i) Find the conditional distribution of X, given Y=1.
(ii) Are X and Y independent. [5+5]
OR
 The input to a binary communication channel is a random variable X taking two
values -1 and +1 with equal likelihood. The output of the system is also a random
variable Y taking values -1,0 and +1. It is given that
P(Y=-1/X=+1)=0.1;P(Y=+1/X=-1)=0.2; P(Y=0/X=+1)=P(Y=0/X=+1)=0.05.
i) When a message is transmitted, what is the probability for it to be received as 0.
ii) When a 0 is received, what is the probability for the transmitted to be a 1.
b) If X and Y are two independent variables, verify that their sum and difference are
of same variance. [6+4]

8.a) Let X(t) be a random process with mean 3 and auto correlation 9+4.exp(-0.2|τ|).
Find the mean, variance and covariance of the random variables X(5) and X(8).
b) Check random process X(t)=A.cos(wt+β), where β is a uniform random variable
over (0, 2π) for mean ergodicity. [5+5]
OR
9.a) A random process is defined as X(t)=ACos(100t+β), where A is a normal random
variable with zero mean and unity variance. β is uniform random variable over (-
π, π) and is independent of A. Find the Autocorrelation function of X(t).
b) Which of the following are valid Autocorrelation functions? Justify:
(i) A.Coswτ (ii) A.Sinwτ (iii) A[u(t+τ)-u(t-τ)] (iv) Triangular pulse from t = -τ to
t= +τ. [5+5]

10.a) A random process with psd of K watts/Hz is applied to an RC LPF with 3dB-
cutoff frequency of fc. Find the power at the output of the filter.
b) A random process X(t) has an autocorrelation function A2  B.e , where A and
B are positive constants. Find the mean of the output of a system with unit
impulse response exp(-kt).u(t), where ‘k’ is a real positive constant, when driven
by X(t). [5+5]
OR
11.a) Two systems with identical unit impulse response of t.exp(- kt)u(t) are in cascade.
If the cascade is driven by a WSS process with a mean of 2, find the mean of the
output of the cascade.
b) A random process X(t) with Autocorrelation function P.exp(-0.2|τ|) is applied to
an LTI system with unit impulse response of K.exp(-Kt).u(t). Find the
Autocorrelation of the response of the system. [4+6]

---ooOoo---
1.a) State and prove Bayes theorem of probability.
b) Find the probability of the card being either red or a king when one card is drawn from a
regular deck of 52 cards. [8+7]

2.a) Given P(A) =1/3, P(B) =1/2, P(A∩B) = 1/5, then find P(AUB) and P(Ac ∩ Bc).
b) Differentiate between the joint probability and conditional probability. [8+7]

The mean of Binomial distribution is 3 and the variance is 9/4. Find

i) The value of n ii) P(x > 7)
b) Describe the properties of moment generating function. [8+7]

4. Mention about the characteristics of the Gaussian and Rayleigh density Functions and
with their distributions graphs in detail. [15]

5. Mention about the Statistical Independence and Show that the variance of a weighted
sum of uncorrelated random variables equal the weighted sum of the Variances of the
random Variables. [15]

6.a) Discuss the Joint Central Moments .

b) If X and Y are two random variables which are Gaussian. If a random Variable Z is
defined as Z=X+Y, Find fZ(Z). [7+8]

7.a) Distinguish between the Ergodic and Mean-Ergodic Processes.

b) Show that PSD and auto correlation function of random process form a Fourier transform
pair. [7+8]

8. A WSS random process x(t) is applied to the input of an LTI system whose impulse
response is 5t.e-2t. The mean of x(t) is 3. Find the output of the system. [15]

---ooOoo---
1.a) A binary communication system transmits a 0 and 1 with equal probabilities. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 0.25 and a
transmitted 0 is received as 1 with a probability of 0.125. (i) If a message received, find
the probability for it to be a 1. (ii) If a zero is received, find the probability that the
transmitted is a 0.
b) What is the probability that a 6 is obtained on one of the dice in a throw of two dice, if
the sum of the two faces is 7. [9+6]

2.a) Find the probability of getting sum of the two faces equal to four, atleast once in three
tosses of a pair of fair dice.
b) A box X contains 3 white and 2 black balls. Box Y contains 2 white and 4 black balls. If
one bag is selected at random and a ball is selected from it, find the probability that the
ball is black.
c) A binary communication source X transmits a 0 and 1, such that P(X=0)=1/7. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 3/8 and a
transmitted 0 is received as a 1 with a probability of 1/4. Find the average probability of
error in the communication system. [5+5+5]

x2
3.a) The probability function of a random variable X is given by f(x) = for -3<x<6 , and
81
1
equal to zero otherwise. Find the density of the random variable Y  12  X  .
3
b) Find the Moment generating function of a random variable, uniform over (a, b) and find
its first two moments about origin, from its Moment Generating function. [7+8]

4.a) Find the Coefficient of skewness of a random variable, uniform over (4,6).
b) What is difference between first order stationary, second order stationary and wide sense
stationary. [7+8]

5.a) The joint density function of two continuous random variables X and Y is given as
f(x,y)=2 for 0<x<1,0<y<x; and is zero elsewhere. Find the marginal densities of X and Y.
b) X and Y are two independent random variables with the individual densities
f  x  0.25  x  0.5  x  2  0.25  x  3; f  y   0.5  x 1  0.5  x  2
Find the density of their sum. [9+6]
6.a) The joint probability matrix of two random variables X and Y is given as
X Y 1 2
0 1 8 1 16
P  x, y  1 1 16 1 16

2 3 16 1 8
3 18 14

Find the conditional distribution of X, given Y=1. Check X and Y for independence.
b) If X and Y are two independent random variables with uniform density over (-1,1) and
(-2,1) respectively, find the P(X + Y ≤ -2). [8+7]

7.a) A random Process is defined as X(t)=3.Cos(2πt+Y), where Y is a random Variable with

p(Y=0)=p(Y=π)=1/2. Find the mean and Variance of the Random Variable X(2).
b) If X(t) and Y(t) are two independent Stationary Processes, find the Auto correlation
function of the process Z(t)=2.X(t).Y(t). [7+8]

8. A stationary random process X(t) has spectral density Sxx(w)=25/(w2+25) and an

independent stationary process Y(t) has the spectral density S yy(w)= w2/(w2+25). If X(t)
and Y(t) are of zero mean, find the (a)PSD of Z(t)=X(t)+Y(t) (b) cross spectral density of
X(t) and Z(t). State and prove the results used. [15]

---oo0oo---
 PART - A
(25 Marks)
1.a) Write the conditions for a function to be a random variable. [2]
b) Explain the significance of mathematical model of experiments. [3]
c) Write short notes on Chebychev’s inequality. [2]
d) Define Characteristic function and present generation of moments using it. [3]
e) State central limit theorem for the case of equal distributions. [2]
f) Write the properties of jointly Gaussian random variables. [3]
g) What is a WSS random process? [2]
h) Write short notes on Gaussian random process. [3]
i) Write the expression for power spectral density. [2]
j) Write any three properties of cross-power density spectrum. [3]

PART - B
(50 Marks)

2. A missile can be accidentally launched if two relays A and B both have failed. The
probabilities of A and B failing are known to be 0.01 and 0.03, respectively. It is
also known that B is more likely to fail (probability 0.06), if A has failed.
a) What is the probability of an accidental missile launch?
b) What is the probability that A will fail, if B has failed?
c) Are the events “A fails” and “B fails” statistically independent? [10]
OR
3. You (A) and two others (B and C) each toss a fair coin in a two-step gambling
game. In step1 the person whose toss is not a match to either of other two is “odd
man out”. Only the remaining two whose coins match go on to step2 to resolve the
ultimate winner.
a) What is the probability that you will advance to step2 after the first toss?
b) What is the probability you will be out after the first toss?
c) What is the probability that no one will be out after the first toss? [10]

4.a) Obtain the moment generating function of a uniformly distributed random variable.
b) Obtain the variance of Raleigh random variable. [5+5]
OR
5.a) A random variable X uniformly distributed in the interval (0, π/2). Consider the
transformation Y=sinx, obtain the pdf of Y.
 b) Obtain the variance of Gaussian random variable. [5+5]

6.a) The joint characteristic function of two random variables is given by

XY(ω1,ω2) = exp(-ω 12- 4ω 22) . Check whether X and Y are uncorrelated or not.
b) X and Y are statistically independent random variables and W = X+Y obtain the pdf
of W. [5+5]
OR
7.a) Write the properties of joint distribution function.
b) Prove that the variance of weighted sum of N random variables equals the weighted
sum of all their covariances. [5+5]

8. Define autocorrelation function of a random process. Write properties of auto

correlation function of a WSS process and prove any three of them. [10]
OR
9.a) A random process X(t) = Acos(ωot) + B sin(ωot) where ωo is a constant and A, B
are uncorrelated zero mean random variables with same variances. Check whether
X(t) is WSS or not?
b) Classify random processes and explain. [5+5]

10. Derive the relationship between cross-power spectrum and cross-correlation

function. [10]
OR
11.a) The autocorrelation function of a random process R XX(τ) = 4 cos(ωoτ), where ωo is a
constant. Obtain its power spectral density.
b) Obtain the average power in the random process X(t) = Acos(ω ot+) where A, ωo
are real constants and  is a random variable uniformly distributed in the range
(0, 2π). [5+5]

--ooOoo--
1.a) Define and explain the concepts of Joint and Conditional probability.
b) How do you explain statistically independent events using Baye’s rule?
c) A bag contains four balls. Two balls are drawn and are found to be white. Find the
probability that all the balls are white. [5+5+5]

2.a) Define conditional distribution and density function of two random variables X and Y.
b) State and prove any three properties of moment generating function. [7+8]

3.a) Define and explain conditional probability mass function. Give its properties.
b) The joint distribution of X and Y is given by
 x 2  y 2 
f  x   4xye , x  0, y  0
Show that X and Y are independent random variables. [7+8]

4.a) X(t) is a stationary random process with a mean of 3 and an auto correlation function of
6+5 exp(-0.2 |τ|). Find the second central Moment of the random variable Y = Z-W,
where Z and W are the samples of the random process at t = 4 sec and t = 8 sec
respectively.
b) Find the cross correlation between the processes X(t) = Acos(ωt) + Bsin(ωt) and
Y(t) = Bcos(ωt) - Asin(ωt), Where A and B are two standardized Gaussian random
variables. [8+7]

5. A stationery random process X(t) has spectral density SXX(ω) = 25/(ω2+25) and an
independent stationery process Y(t) has the spectral density S YY(ω) = ω2/(ω2+25). If X(t)
and Y(t) are zero mean, find the:
a) PSD of Z(t) = X(t) + Y(t)
b) Cross spectral density of X(t) and Z(t). State and prove the results used. [7+8]

6.a) Distinguish between ensemble average and time average of a random process.
b) A random process is defined as X(t) = A sin(ωt + θ), where A is a constant and θ is a
random variable uniformly distributed over (-π, π). Check X(t) for stationarity. [7+8]

7.a) For two jointly stationary random processes, the cross-correlation function is
RXY(τ) = 2e-2τ u(τ). Find the two cross-spectral density functions.
b) List the properties of cross power spectral density function. [7+8]

8.a) Show that the autocorrelation function of a stationary random process is an even function
of τ.
b) Discuss the properties of conditional distribution function. [7+8]

---ooOoo---
 PART- A
(25 Marks)
1.a) When two dice are thrown simultaneously, if X and Y denote the numbers on the first
and second respectively, find the probability for X+Y to be greater than or equal to 8.
[2]
b) A box contains three coins: one is fair, one is two headed and one coin is weighted so that
the probability of head is 1/3. A coin is selected at random and tossed. Find the
probability for head to appear. [3]
c) A random variable X is having a CDF as shown:

Find the density of X over( -2, 1) and P(X=1.2). [2]

 x for 0  x  1
d) The density function of a random variable X is f x 2  x for 1  x  2 . Find its
 
 for x  2 
 0
CDF. [3]
e) A fair coin is tossed three times. Let X denote a „0‟ or „1‟ according a head or tail occurs
in the first toss and let Y denote the number of heads which occur. Write the joint
probability Matrix of X and Y. [2]
f) The joint density of two random variables X and Y is given as

f x, y  0.21  x 1  y  2  0.35 x 1  y  5  0.14 x 1  y  8  0.09 x  2  y

 2
0.15  x  2  y  5  0.06  x  2  y  8
Find P(Y is even, given X is even). [3]
g) State and prove the Periodicity property of Auto correlation function of a periodic WSS
process. [2]
 h) If X(t) and Y(t) are two independent Stationary Processes, find the Auto correlation
function of the process Z(t)=2.X(t).Y(t). [3]
i) Verify that Cross Spectral Density of two Uncorrelated and Stationary Random processes
is an impulse function. [2]
j) The input to an LTI system with Impulse response h(t)=δ(t)+t2.exp(-at).U(t) is a WSS
process with mean of 3.Find the mean of the output of the system. [3]

PART-B
(50 Marks)
2.a) Three machines A, B and C produce 55%, 25% and 20% of the total number of items of a
factory. The percentage of defective output of these machines are 3%, 2% and 4%. If an
item is selected at random, (i) find the probability that the item is defective
(ii) If the selected item is defective, find the probability that the item is produced by
Machine B.
b) In a single throw of a pair of dice, what is the probability of obtaining the sum of two
faces of the dice is equal to at least 10. [6+4]
OR
3.a) Two different digits are selected at random from the digits 1 to 9. (i) If the sum of the
digits is odd, what is the probability that 2 is one of the digits selected? (ii) If 2 is one of
the selected digits, what is the probability for the sum to odd.
b) A binary communication system transmits a 0 and 1 with equal probabilities. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 1/8 and a
transmitted 0 is received as 1 with a probability of 3/4. (i) Find the probability for the
transmitted message to be a 1. (ii) If a one is received, find the probability that the
transmitted is a 1. [6+4]

4.a) A discrete random variable is having a density function given by

f(x)=0.1δ(x)+0.4δ(x-1)+0.3δ(x-2)+0.2 δ(x-3).
x3
Find the density of the random variable Y  2  x2  .
3
b) Verify that Gaussian Density function is a valid density function. [5+5]
OR
5.a) Find the Mean and Mean square value of Rayleigh Random Variable.
b) X is a Gaussian random variable with zero mean and unity variance. Find the density of
the random variable Y = X for x ≥ 0, and Y=0 for x < 0. [5+5]
5
6.a) Two random variables X and Y have a joint probability density function f(x,y)= 2
16 x y
for 0 < y < x < 2; and zero elsewhere. Check X and Y for independence.
b) Let f(x,y) = x + y for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; and zero elsewhere. Find the density of X,
given Y. [6+4]
OR
7.a) The joint probability mass function of two random variables X and Y is
P(x,y)=K(2x+3y) for x=0,1,2; and y=1,2,3. Find the marginal probability distributions of
X and Y. Find all the possible conditional probabilities of X.
b) If X is a random variable with mean 3 and variance 2, verify that the random variables
„X‟ and Y=-6X+22 are orthogonal. [6+4]
8.a) A Random Process X(t)=A.Cos (2π fc t) , where A is a Gaussian Random Variable with
zero mean and unity variance, is applied to an ideal integrator, that integrates with respect
to „t‟, over (0,t). Check the output of the integrator for stationarity.
b) In the random Process X(t)=A.CosWt, A is uniform random Variable over (0,1) and W is
a constant. Find the Auto correlation function of X(t). [6+4]
OR
9.a) X(t) is a random process with mean =3 and Autocorrelation function
Rxx(τ) =10.[exp(- 0.3|τ|)+2]. Find the second central Moment of the random variable
Y=X(3)-X(5).
b) X(t) is a WSS process and Y(t)=A.Cos(Wct+θ) is a random process which is independent
of X(t). Here, θ is a uniform random variable over(-π,π).If the Auto correlation function
of X(t) is Rxx(τ), find the Auto correlation of Z(t)=X(t).Y(t). [5+5]

10.a) Find the Auto correlation and PSD and M.S. Value of the random process, X(t)= m(t).
Cos(Wt+Ф), where m(t) is a WSS process and „Ф‟ is a uniform random variable over
(0,2π), and is independent of m(t).
b) A noise process with zero mean and of PSD “K” is applied to an R-L LPF. Find the Mean
Square value of the output Process. [5+5]
OR
11.a) Let x(t)=Y(t) -Y(t-2) is a random process, where Y(t) is also a stationary random process.
It is given Var(X(t))=20.Var(Y(t)). Find RYY(2)/var(X(t)) and also BXX.
b) X(t)=A.Sin(wt+θ) is a random process, with „θ‟ being a uniform random variable over
the interval (-π, π). If Y(t)=(1/2).X(t). Are X(t) and Y(t) are jointly stationary and find
Syy.
[5+5]

---oo0oo---
 PART- A
(25 Marks)
1.a) A box contains nine cards numbered through 1 to 9, and B contains five cards
numbered through 1 to 5. If a box is chosen at random, and a card is drawn which
even numbered, what is the probability for the card to be from box A. [2]
b) Let a die be weighted such that the probability of getting numbers from 2 to 6 is
that number of times of probability of getting a1. When the die thrown, what is
the probability of getting an even or prime number occurs. [3]
c) Find the CDF of a random variable X, uniform over (-3, 3). [2]
d) The density of a random variable X is given as f(x)= K[U(x)-U(x-4)]+0.25δ(x-2).
Find the probability of X ≤ 3. [3]
e) X and Y are discrete random variables and their joint occurrence is given as
X\Y 1 2 3
1 1/18 1/9 1/6
2 1/9 1/18 1/9
3 1/6 1/6 1/18
Find the Conditional Mean of X, given Y=2. [2]
f) X and Y are two uncorrelated random variables with same variance. If the random
variables U=X+ kY and V=X+(σx/σy)Y are uncorrelated, find K. [3]
g) State and prove the Periodicity Property of Auto Correlation function of a
Stationary Random Process. [2]
h) If X(t) is a Gaussian Random Process with a mean 2 and exp (-0.2|τ|). Find the
Probability of X(1) ≤ 1. [3]
i) Verify that the cross spectral density of two uncorrelated stationary random
processes is an impulse function. [2]
j) The output of a filter is given by Y(t)=X(t+T)+X(t-T), where X(t) is a WSS
process, power spectral density Sxx(w), and T is a constant. Find the power
spectrum of Y(t). [3]
PART-B
(50 Marks)

2.a) Consider the experiment of tossing two dice simultaneously. If X denotes the sum
of two faces, find the probability for X ≤ 6.
b) A fair coin is tossed 4 times. Find the probability for the longest string of heads
appearing to be three as a result of the above experiment.
c) In certain college, 25% of the boys and 10% of the girls are studying
Mathematics. The girls constitute 60% of the student body. If a student is
selected at random and studying mathematics, determine the probability that the
student is a girl. [3+3+4]
OR
3.a) Coin A has a probability of head =1/4 and coin B is a fair coin. Each coin is
flipped four times. If X is the number of heads resulting from coin and Y denotes
the same from coin B, what is the probability for X=Y?
b) A dice is thrown 6 times. Find the probability that a face 3 will occur at least two
times. [6+4]

4.a) Find the Moment generating function of a uniform random variable distribute
over (A, B) and find its first and second moments about origin, from the Moment
generating function.
b) A random variable X has a mean of 10 and variance of 9. Find the lower bound on
the probability of (5<X<15). [5+5]
OR
5.a) Find the Moment generating function of a random variable X with density
function
 x, for 0  x  1 
f  x  2  x, for 1  x  2
 
 
 0, else where 
b) If X is a Gaussian random variable N(m, σ2 ), find the density of Y=PX+Q, where
P and Q are constants. [5+5]

6.a) If X1,X2,X3, - - - - - Xn are ‘n’ number of independent and Identically distributed

random variables, such that Xk = 1 with a probability 1/2; = -1 with a probability
1/2. Find the Characteristic Function of the random Variable Y= X1+X2+X3+ - - -
+ Xn.
b) If Independent Random Variables X and Y both of zero mean, have variance 20
and 8 respectively, find the correlation coefficient between the random Variables
X+Y and X-Y. [5+5]
OR
7.a) Let X=Cosθ and Y=Sinθ , be two random variables, where θ is also a uniform
random variable over (0,2π). Show that X and Y are uncorrelated and not
independent.
b) If X is a random variable with mean 3 and variance 2, verify that the random
Variables ‘X’ and Y= -6X+22 are orthogonal. [6+4]

8.a) X(t) is a random process with mean =3 and Autocorrelation function

Rxx(τ) =10.[exp(- 0.3|τ|)+2]. Find the second central Moment of the random
variable Y=X(3)-X(5).
b) X(t)=2ACos(Wct+2θ) is a random Process, where ‘θ’ is a uniform random
variable, over (0,2π). Check the process for mean ergodicity. [5+5]
OR
9.a) A Random Process X(t)=A.Cos (2π fc t) , where A is a Gaussian Random
Variable with zero mean and unity variance, is applied to an ideal integrator, that
integrates with respect to ‘t’, over (0,t). Check the output of the integrator for
stationarity.
b) A random Process is defined as X(t)=3.Cos(2πt+Y), where Y is a random
Variable with p(Y=0)=p(Y=π)=1/2.Find the mean and Variance of the Random
Variable X(2). [5+5]
10.a) Find and plot the Autocorrelation function of (i) Wide band white noise
(ii) Band Pass White noise.
b) Derive the expression for the Cross Spectral Density of the input Process X(t)
and the output process Y(t) of an LTI system in terms of its Transfer function.
[5+5]
OR
11.a) Compare and contrast Auto and cross correlations.
b) If Y(t) = A.Cos(w0t+θ)+N(t), where ‘θ’ is a uniform random variable over (-π,π),
and N(t) is a band limited Gaussian white noise process with PSD=K/2. If ‘θ’ and
N(t) are independent, find the PSD of Y(t). [4+6]

---ooOoo---
 PART- A
(25 Marks)
1.a) The joint occurrence of two events A and B is given as
 A B 1 2 3 
1 1 12 1 6 1 12
P  A, B    . Find the probability of B is even, given that
 2 1 6 1 4 1 12
3 1 12 1 12 0
  
A is even. [2]
b) A die is weighted so that even numbers have same chance of appearing, and the
odd numbers have the same chance of appearing, and each even number is twice
as likely to appear as any odd number. When the die is tossed, what is the
probability of getting an even numbered face. [3]
c) A random variable X has a CDF given by
 0; for x  0

F  x   cos x . Find  x  2  , and justify the answer. [2]
X 1  0x
; for
FX

 2
d) A random variable X has the following distribution:

xi 0 1 2 3 4 5 6 7 8
P(xi) a 3a 5a 7a 9a 11a 13a 15a 17a

Find the smallest value of x, for which P[(X ≤ x) > 0.5]. [3]
e) A fair coin is tossed three times. Let X denote a ‘0’ or a ‘1’ according a head or
tail occurs in first toss and let Y denote the number of heads which occur. Find
the joint distribution of X and Y. [2]
f) Find the correlation coefficient between random variables X and Y related as
X=20Y. [3]
g) A random process is defined as X(t)= A.Coswt, where A is a uniform random
variable. What is the condition on A for the process to be WSS? [2]
h) X(t)=A Cos(2πt+Y) is a random process, where Y is a random variable such that
P(Y=0)=1/2 and P(Y=π/2)= 1/2. Find the correlation between the random
variables X(0) and X(1). [3]
i) A random process with PSD of K watts/Hz is applied to an ideal LPF with pass
band over (-B Hz to +B Hz). Find the noise power at the output of the filter. [2]
j) Find the cross spectral density of two uncorrelated stationary random processes
X(t) and Y(t). [3]
 PART-B
(50 Marks)
2.a) A coin is flipped three times and X denotes the number of heads that show up.
The probability of getting a head in each flip is ‘q’. Give the probability
distribution of X. Find the probability with which X>1.
b) A pair of fair dice is thrown. Find the probability that the sum is 10 or greater if
(i) 5 appears on the first dice (ii) 5 appears at least on one of the dice. [5+5]
OR
3.a) Find the probability of getting a total of 4 at least once in 3 tosses of a pair of
dice.
b) Two different digits are selected at random from digits 1 to 9. If the sum of the
digits selected is odd, what is the probability for 2 to be one of the numbers
selected?
c) A pair of fair dice is tossed. Find the probability that the maximum of the two
numbers is greater than four. [3+4+3]

4.a) The pdf of a random variable X is given by

f  x   K u  x   u  x  6  0.25  x  3  0.25  x  6 . Find and plot
the CDF of X.
b) The density function of a random variable X is given as
 ax; for 0  x  1 
 a; for 1  x  2 
f  x    . If three independent observations are made on
 3a  ax; for 2  x  3 
 0; else where 
X, find the probability with which exactly one of them is greater than 1.5. [5+5]
OR
5.a) Find the density of the Random variable Y=Tan(X), where X is uniform random
variable over ( -π/2, π/2).
b) For a random variable X with pdf f(x)= 0.5 exp(-|x|), find the probability that
an observed value will fall within ±2σ2 of the mean value. [4+6]

6.a) Find the density of Z=X+Y, where X and Y are two independent random
variables, which are uniform over (-2, +2) and (-1, 3) respectively.
b) A product is classified according to the number of defects it contains (X) and the
factory (Y) producing the product. The joint probability distribution is given by
 X Y 1 1 
0 1 8 1 16
 
 1 1 16 1 16
2 3 16 1 8
 
 3 1 8 1 4 
(i) Find the conditional distribution of X, given Y=1.
(ii) Are X and Y independent. [5+5]
OR
 The input to a binary communication channel is a random variable X taking two
values -1 and +1 with equal likelihood. The output of the system is also a random
variable Y taking values -1,0 and +1. It is given that
P(Y=-1/X=+1)=0.1;P(Y=+1/X=-1)=0.2; P(Y=0/X=+1)=P(Y=0/X=+1)=0.05.
i) When a message is transmitted, what is the probability for it to be received as 0.
ii) When a 0 is received, what is the probability for the transmitted to be a 1.
b) If X and Y are two independent variables, verify that their sum and difference are
of same variance. [6+4]

8.a) Let X(t) be a random process with mean 3 and auto correlation 9+4.exp(-0.2|τ|).
Find the mean, variance and covariance of the random variables X(5) and X(8).
b) Check random process X(t)=A.cos(wt+β), where β is a uniform random variable
over (0, 2π) for mean ergodicity. [5+5]
OR
9.a) A random process is defined as X(t)=ACos(100t+β), where A is a normal random
variable with zero mean and unity variance. β is uniform random variable over (-
π, π) and is independent of A. Find the Autocorrelation function of X(t).
b) Which of the following are valid Autocorrelation functions? Justify:
(i) A.Coswτ (ii) A.Sinwτ (iii) A[u(t+τ)-u(t-τ)] (iv) Triangular pulse from t = -τ to
t= +τ. [5+5]

10.a) A random process with psd of K watts/Hz is applied to an RC LPF with 3dB-
cutoff frequency of fc. Find the power at the output of the filter.
b) A random process X(t) has an autocorrelation function A2  B.e , where A and
B are positive constants. Find the mean of the output of a system with unit
impulse response exp(-kt).u(t), where ‘k’ is a real positive constant, when driven
by X(t). [5+5]
OR
11.a) Two systems with identical unit impulse response of t.exp(- kt)u(t) are in cascade.
If the cascade is driven by a WSS process with a mean of 2, find the mean of the
output of the cascade.
b) A random process X(t) with Autocorrelation function P.exp(-0.2|τ|) is applied to
an LTI system with unit impulse response of K.exp(-Kt).u(t). Find the
Autocorrelation of the response of the system. [4+6]

---ooOoo---
1.a) State and prove Bayes theorem of probability.
b) Find the probability of the card being either red or a king when one card is drawn from a
regular deck of 52 cards. [8+7]

2.a) Given P(A) =1/3, P(B) =1/2, P(A∩B) = 1/5, then find P(AUB) and P(Ac ∩ Bc).
b) Differentiate between the joint probability and conditional probability. [8+7]

The mean of Binomial distribution is 3 and the variance is 9/4. Find

i) The value of n ii) P(x > 7)
b) Describe the properties of moment generating function. [8+7]

4. Mention about the characteristics of the Gaussian and Rayleigh density Functions and
with their distributions graphs in detail. [15]

5. Mention about the Statistical Independence and Show that the variance of a weighted
sum of uncorrelated random variables equal the weighted sum of the Variances of the
random Variables. [15]

6.a) Discuss the Joint Central Moments .

b) If X and Y are two random variables which are Gaussian. If a random Variable Z is
defined as Z=X+Y, Find fZ(Z). [7+8]

7.a) Distinguish between the Ergodic and Mean-Ergodic Processes.

b) Show that PSD and auto correlation function of random process form a Fourier transform
pair. [7+8]

8. A WSS random process x(t) is applied to the input of an LTI system whose impulse
response is 5t.e-2t. The mean of x(t) is 3. Find the output of the system. [15]

---ooOoo---
1.a) A binary communication system transmits a 0 and 1 with equal probabilities. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 0.25 and a
transmitted 0 is received as 1 with a probability of 0.125. (i) If a message received, find
the probability for it to be a 1. (ii) If a zero is received, find the probability that the
transmitted is a 0.
b) What is the probability that a 6 is obtained on one of the dice in a throw of two dice, if
the sum of the two faces is 7. [9+6]

2.a) Find the probability of getting sum of the two faces equal to four, atleast once in three
tosses of a pair of fair dice.
b) A box X contains 3 white and 2 black balls. Box Y contains 2 white and 4 black balls. If
one bag is selected at random and a ball is selected from it, find the probability that the
ball is black.
c) A binary communication source X transmits a 0 and 1, such that P(X=0)=1/7. Due to the
noise in the channel, a transmitted 1 is received as a 0 with a probability of 3/8 and a
transmitted 0 is received as a 1 with a probability of 1/4. Find the average probability of
error in the communication system. [5+5+5]

x2
3.a) The probability function of a random variable X is given by f(x) = for -3<x<6 , and
81
1
equal to zero otherwise. Find the density of the random variable Y  12  X  .
3
b) Find the Moment generating function of a random variable, uniform over (a, b) and find
its first two moments about origin, from its Moment Generating function. [7+8]

4.a) Find the Coefficient of skewness of a random variable, uniform over (4,6).
b) What is difference between first order stationary, second order stationary and wide sense
stationary. [7+8]

5.a) The joint density function of two continuous random variables X and Y is given as
f(x,y)=2 for 0<x<1,0<y<x; and is zero elsewhere. Find the marginal densities of X and Y.
b) X and Y are two independent random variables with the individual densities
f  x  0.25  x  0.5  x  2  0.25  x  3; f  y   0.5  x 1  0.5  x  2
Find the density of their sum. [9+6]
6.a) The joint probability matrix of two random variables X and Y is given as
X Y 1 2
0 1 8 1 16
P  x, y  1 1 16 1 16

2 3 16 1 8
3 18 14

Find the conditional distribution of X, given Y=1. Check X and Y for independence.
b) If X and Y are two independent random variables with uniform density over (-1,1) and
(-2,1) respectively, find the P(X + Y ≤ -2). [8+7]

7.a) A random Process is defined as X(t)=3.Cos(2πt+Y), where Y is a random Variable with

p(Y=0)=p(Y=π)=1/2. Find the mean and Variance of the Random Variable X(2).
b) If X(t) and Y(t) are two independent Stationary Processes, find the Auto correlation
function of the process Z(t)=2.X(t).Y(t). [7+8]

8. A stationary random process X(t) has spectral density Sxx(w)=25/(w2+25) and an

independent stationary process Y(t) has the spectral density S yy(w)= w2/(w2+25). If X(t)
and Y(t) are of zero mean, find the (a)PSD of Z(t)=X(t)+Y(t) (b) cross spectral density of
X(t) and Z(t). State and prove the results used. [15]

---oo0oo---
1.a) In the experiment of tossing a dice, what is the probability of face having 3 dots or 6 dots
to appear?
b) Define sample space and classify the types of sample space.
c) In the experiment of tossing a die, all the even numbers are equally likely to appear and
similarly the odd numbers. An odd number occurs thrice more frequently than an even
number. Find the probability that:
i) an even number appears
ii) a prime number appears
iii) an odd numbers appears
iv) an odd prime number appears. [4+4+7]

2.a) Derive the Binomial density function and find mean and variance.
b) The probability density function of a random variable X is given by f(x) =x2/81
for -3<x<6 and equal to zero otherwise. Find the density function of Y=0.33(12-x). [7+8]

3.a) State and prove the chebychev’s inequality theorem.

b) Let x be the random Variable with probability law P(X = r) = qr−1p, r = 1, 2, 3......Find the
moment generating function and hence mean and Variance, Assume p+q=1. [8+7]

4.a) The joint density function of random variables X and Y is

8xy; 0  x  1, 0  y  1
f xy  x, y  
 0; otherwise
Find f(y/x) and f(x/y). Find the variances of X and Y.
b) Define and explain joint distribution function and joint density function of two random
variables X and Y. [8+7]

5.a) Show that the variance of a weighted sum of uncorrected random variables equals the
weighted sum of the variances of the random variables.
b) For two zero mean Gaussian random variables X and Y show that their joint
characteristic function is φXY(ω1,ω2) = exp{-1/2[σX2ω12+2ρσXσYω1ω2+σY2ω22]}.
[7+8]
Explain the following:
i) Stationarity
ii) Ergodicity
iii) Statistical independence with respect to random processes.
b) Explain the classification of random processes with neat sketches. [8+7]
7.a) Check the following power spectral density functions are valid or not
−1)2
𝑖) 𝑐𝑜𝑠8(𝜔)2 + 𝜔4 𝑖𝑖) 𝑒−(𝜔
b) Derive the relation between input PSD and output PSD of an LTI system. [8+7]

8.a) What are the important parameters that determine the overall noise figure of a multistage
filtering?
b) Describe the quadrature representation of narrowband noise. [8+7]
 1.a) Discuss Joint and conditional probability.
b) When are two events said to be mutually exclusive? Explain with an example. [8+7]

2.a) State and prove any four properties of distribution function.

b) Derive expressions for mean and variance for binomial random variable. [7+8]

3.a) Find the expected value of uniformly distributed random variable.

b) If X be a discrete random variable with probability mass function is given as
X -2 -1 0 1 2
P(x) 0.2 0.4 0.1 0.1 0.2

2 2
Find: i) E[X ] ii) E[(2X+1) ]. [7+8]

4.a) Define and explain the properties of conditional density function.

b) Joint probabilities of two random variables X and Y are given in figure.

X
1 2 3
Y
1 0.2 0.1 0.2
2 0.15 0.2 0.15
Find out:
i) Joint and marginal distribution functions.
ii) Joint and marginal density functions. [7+8]

5.a) Prove that the joint characteristic functions of two independent random variables X and Y
is equal to the product of their individual characteristic functions.
b) Discuss the properties of Gaussian random variables. [8+7]

6.a) Find the mean and auto correlation function of a random process X(t)=A, where A is
continuous random variable with uniform distribution over (0,1).
b) What are the conditions for a Wide sense stationary random process? Explain. [8+7]
7.a) The cross Power spectral density is given as

Find out the cross correlation function.

b) Prove that SXY(ω) = 0 and SYX(ω) = 0, If X(t) and Y(t) are orthogonal. [8+7]

Explain:
i) Transit – time noise ii) Thermal noise
b) Show that for RC low pass filter shown in figure, the noise bandwidth is equal to π/2
times of 3-dB bandwidth. [8+7]

---ooOoo---
 ---

1.a) A verification plan calls for verification of five chips and for either accepting each chip,
rejecting each chip, or submitting it for reverification, with probabilities of p1= 0.70,
p2= 0.20, p3= 0.10 respectively. What is the probability that all five chips must be
reverified? What is the probability that none of the chips must be reverified?
b) State and prove Baye’s theorem. [8+7]

2.a) Define conditional density and write its properties. Also discuss different methods of
defining the conditioning event and obtain the respective density functions.
b) Define probability density function and write its properties. [8+7]

3.a) Obtain mean of uniformly distributed random variable between (1, 5).
b) Define characteristic function of a random variable and explain how moments can be
generated using it. [7+8]

4.a) Define joint density function of two random variables and write its properties.
b) The joint pdf ofxyrandom variables X and Y is given
, 0  x  2 and 0  y  3
f X ,Y  x, y   

9 else where
 0,
Check whether X and Y are correlated or not. [7+8]

5.a) Two random variables X and Y have joint characteristic function

Show that X and Y are zero mean random variables and uncorrelated.
b) The density function of two random variables X and Y is

.
Find the mean value of the function e–(X+Y). [7+8]

6.a) Classify random processes and explain.

b) Check whether the random process X(t) = A cos(ωot +Θ) is WSS process or not, for ‘A’
and ωo being constant and Θ uniformly distributed between (0, 2π). [7+8]

7.a) Define autocorrelation function of a random process and write its properties.
b) Derive the expression for power density spectrum of a random process. [8+7]

8. Write short notes on:

a) Extra terrestrial Noise
b) Resistive Noise
c) Properties of narrow band noise. [5+5+5]

---ooOoo---
1.a) Explain how the concept of probability can be applied in communication system.
b) State and prove the total probability theorem.
c) Calculate correct and error transmission probabilities of binary symmetric channel
using Baye’s theorem by assuming your own values. [5+5+5]

2.a) Explain the applications of all types of continuous and discrete random variables.
b) Find the probability of getting sum in random experiment of rolling two dice.
Find, plot and obtain the expression for both PDF and CDF? [7+8]

3.a) Obtain the expression for all statistical parameters and explain the significance in
analyzing communication systems.
b) Define mean, variance and skew of binomial random variable. [8+7]

4.a) State and prove the all properties of joint PDF and CDF.
b) Find the expression for PDF of sum of two independent variables. [7+8]

5.a) State and prove the all properties of correlation and covariance.
b) Find constant ‘C’, correlation and covariance of a two random variables ‘X’ and
‘Y’ having joint PDF
C  2x  y  ; 0  X  2 and 0  Y  3
f xy  x, y  
 Else where

0;
Is ‘X’ and ‘Y’ are independent. [7+8]

6.a) State and prove any ‘THREE’ properties of Auto Correlation Function(ACF).
And also explain their significance.
b) Mean and ACF of random process X(t) is given by 6 RXX    36  25 et
and
i) Is first order stationary ii) Find the total power of X(t)
iii) Is Ergodic iv) Is Wide Sense Stationary
v) Has periodic components vi) Find AC power of X(t). [7+8]

7.a) Derive the relation between input and output PSDs of LTI system.
b) A random process X(t) whose ACF is given by R    4 is applied to a
XX
e2

system with transfer function 1

H   . Find out mean value, ACF, PSD
2  j
and total power of output random process Y(t). [7+8]
8.a) Define noise. Explain different types of noise generated in communication
system.
b) With help of noise figure, explain why noise generated in first stage should be
minimum in cascade amplifiers. [7+8]

---ooOoo---
1.a) State and Proof Baye’s Theorem.
b) A class contains 9 boys and 3 girls.
In how many ways can the teacher choose a committee of 4?
How many of them will contain at least one girl?
How many of them will contain exactly one girl? [7+8]

2.a) Explain in detail the cumulative distribution function. Explain the properties of
cumulative distribution function.
b) Explain in detail the Binomial and Poisson’s Distribution. [7+8]

3.a) Write short notes on Expected Value of Random Variable. Write about Central
Moments.
b) Explain in detail the chebyshev’s Inequality. [7+8]

4.a) Define joint distribution and joint probability density function for the two random
variables X and Y. Distinguish between joint distribution and marginal distribution.
b)
is A joint probability density function 1
f  x, y  
ab, for 0  x  a, 0  y  b


 0, elsewhere
Find the joint probability distribution function. [10+5]

5.a) Write short notes on Joint moments about the origin for two random variables.
b) Two random variables X and Y has the following joint probability distribution
function as
2  x  y for 0  x  1
f  x, y  
 
 0, elsewhere
Find the var(X) and var(Y). [7+8]

6.a) Write short notes on auto correlation function. State all the properties of auto
correlation function.
b) Consider a random variable process X (t) = Acosωt where ‘ω’ is a constant and A is a
random variable uniformly distributed over the interval (0,1). Find the autocorrelation
and covariance of X (t). [7+8]

7.a) State and Proof Wiener-Khintchine relation.

b) Discuss the properties of power spectrum. [10+5]

8. Write short notes on:

a) Shot Noise
b) Thermal Noise
c) White Noise. [5+5+5]

1.a) State and Proof Baye’s Theorem.

b) A class contains 9 boys and 3 girls.
In how many ways can the teacher choose a committee of 4?
How many of them will contain at least one girl?
How many of them will contain exactly one girl? [7+8]

2.a) Explain in detail the cumulative distribution function. Explain the properties of
cumulative distribution function.
b) Explain in detail the Binomial and Poisson’s Distribution. [7+8]

3.a) Write short notes on Expected Value of Random Variable. Write about Central
Moments.
b) Explain in detail the chebyshev’s Inequality. [7+8]

4.a) Define joint distribution and joint probability density function for the two random
variables X and Y. Distinguish between joint distribution and marginal distribution.
b)
is A joint probability density function 1
f  x, y  
ab, for 0  x  a, 0  y  b


 0, elsewhere
Find the joint probability distribution function. [10+5]

5.a) Write short notes on Joint moments about the origin for two random variables.
b) Two random variables X and Y has the following joint probability distribution
function as
2  x  y for 0  x  1
f  x, y  
 0, elsewhere
Find the var(X) and var(Y). [7+8]

6.a) Write short notes on auto correlation function. State all the properties of auto
correlation function.
b) Consider a random variable process X (t) = Acosωt where ‘ω’ is a constant and A is a
random variable uniformly distributed over the interval (0,1). Find the autocorrelation
and covariance of X (t). [7+8]

7.a) State and Proof Wiener-Khintchine relation.

b) Discuss the properties of power spectrum. [10+5]
---ooOoo---
1.a) Define Equally likely events, Exhaustive events and Mutually exclusive events.
b) The six sides of a fair die are numbered from t to 6. The die is rolled 4 times. How many
sequences of the four resulting numbers are possible?
c) In a factory there are 4 machines produce 10%, 20%, 30%, 40% of an items respectively.
The defective items produced by each machine are 5%,4%,3% and 2% respectively. Now
an item is selected which is to be defective, what is the probability it being from the 2nd
machine. And also write the statement of total probability theorem? [4+4+7]

2.a) Derive the Poisson density function and find mean and variance.
b) Let X be a Continuous random variable with density function
x 9  K ; 0  x  6
f  x 
 0; otherwise
Find the value of K and also find P{2 ≤ X ≤ 5} [7+8]

3.a) Verify the Characteristic function of a random variable is having its maximum magnitude
at ω=0 and find its maximum value.
b) If X is a discrete random variable with a Moment generating function of Mx(v), find the
Moment generating function of
i) Y=aX+b ii)Y=KX iii) Y=𝑋+𝑎/𝑏 [7+8]

4.a) State and explain the properties of joint density function.

b) Joint probabilities of two random variables X and Y are given in Figure.

Find out:
i) Joint and marginal distribution functions and plot.
ii) Joint and marginal density functions and plot. [8+7]
5.a) Let X and Y be independent random variables each N(0,1). Find the mean and variance
of Z =𝑋2+𝑌2.
b) Random variables X and Y have the joint density function
fX,Y(x, y) = (x + y)2/40; −1 < x < 1 and − 3 < y < 3 find all the third order moments for
X and Y. [7+8]

Explain about the following random process

i) Mean ergodic process
ii) Correlation ergodic process
iii) Gaussian random process.
b) i) Given the RP X(t) = A cos(w0t) + B sin (w0t) where ω0 is a constant, and A and B are
uncorrelated Zero mean random variables having different density functions but the same
variance σ2. Show that X(t) is wide sense stationary.
ii) Define Covariance of the Random processes with any two properties. [7+8]

7.a) The input to an LTI system with impulse response h(t)= 𝛿 (𝑡) + 𝑡2𝑒−𝑎𝑡 . u(t) is a WSS
process with mean of 3. Find the mean of the output of the system.
b) A random process has the power density spectrum SYY(ω)=6 ω2/1+ ω4. Find the average
power in the process. [8+7]

8.a) The noise figure of an amplifier at room temperature (T=29.K) is 0.2dB. Find the
equivalent temperature.
b) Show that the effective noise temperature of n networks in cascade is given by,
Te = Te1 + Te2/g1 + Te3/g1g2 + ................... + Ten/g1g2..gn−1 [8+7]

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Code No: 53019
R09
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
B.Tech II Year I Semester Examinations, September - 2021
PROBABILITY THEORY AND STOCHASTIC PROCESSES
(Electronics and Communication Engineering)
Time: 3 hours Max. Marks: 75
Answer any five questions
All questions carry equal marks
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Two boxes A and B contain 80 and 160 light bulbs respectively. A and B have 10 and 5
defective bulbs respectively.
i) Suppose a box is selected at random and one bulb is picked out. What is the probability
that it is defective?
ii) Suppose we test the bulb and it is found to be defective. What is the probability, which
it came from A?
b) Explain the term independent events and also write the properties of independent events.
[8+7]
2.a) Given c is a constant and X is a random variable with pdf
cx, 0  x  1
f  x 
x
 0, else where
Find the value of c and P[1/2 < x < ¾].
b) Define probability distribution function and write its properties. [8+7]

3.a) Define moment generating function of a random variable and explain how moments
can be generated using it.
b) Find mean of Poisson random variable. [8+7]

4.a) Two random variables X and Y have a joint probability density function

Check whether X and Y are statistically independent or not.

b) Define joint distribution function of two random variables and write its properties. [8+7]

5.a) Consider the transformations Y  aX  bX , Y  cX  where a, b, c, d are real

1 1 2 2 1
dX2
constants and X1 and X2 are random variables. Obtain the expression for fY1Y2(y1,y2).
b) Define joint characteristic function and write its properties. [8+7]

6.a) Given that the autocorrelation function for a stationary ergodic process with no periodic
components 4
R T   25 . Find the mean and variance of the process X(t).
is
 1  6T 2


b) Explain the terms wide sense stationarity and strict sense stationarity. [8+7]

7.a) State and prove Wiener-Khintchine relations.

b) Write properties of power density spectrum. [8+7]

8.a) Obtain the expression for average noise figure of cascaded two port network.
b) [8+7]
Write notes on average noise bandwidth and effective noise temperature.

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