Jawaharlal Nehru Technological University Kakinada 2008 B.Tech Electronics and Communications Engineering PROBABILITY THEORY AND STOCHASTIC PROCESSES - Question Paper

2. (a) elaborate point conditioning and interval conditioning distribution function?
Explain.
(b) If P(x) = 0.1x, x = 1,2,3,4
= 0, otherwise
Find:
i. P{X = one or 2}
ii. P{(1/2) < X (5/2)X > 1}. [8+8]
3. (a) If the random variable X has the moment generating function MX (t) = 2
2-t
,
determine the variance of X.
(b) Show that the distribution function for which the characteristic function e-|t|
has the density:
fX (x) = 1
p(1+x2)
,-8 < x < 8
(c) discuss the nonmonotonic transformation of a random variable. [6+6+4]
4. (a) De?ne and discuss conditional probability mass function. provide its properties.
(b) The joint probability density function of 2 random variables X and Y is
provided by
f(x, y) = C(2x + y), 0 = x = 1, 0 = y = 2
= 0, elsewhere
Find:
i. the value of ‘C’
ii. Marginal distribution functions of X and Y. [8+8]
5. (a) Write the expression for expected value of a function of random variables and
prove that the mean value of a weighted sum of random variables equals the
weighted sum of mean values.
(b) X is a random variable with mean ¯ X = 3, variance s2
X = 2.
i. Determine the 2nd moment of X about origin
1 of 2Code No: 07A3EC10 Set No. 3
ii. Determine the mean of random variable y = where y = -6X +22. [8+8]
6. explain in detail about:
(a) 1st order stationary random process
(b) 2nd order & Wide - Sense Stationary Random Process. [8+8]
7. The auto correlation function of a random process X(t) is RXX (t ) = 3+2 exp (-4t 2
).
(a) obtain the power spectrum of X(t).
(b) What is the avg. power in X(t)
(c) What fractional power lies in the frequency band -1 v2
= ? = one v2
. [6+4+6]
8. (a) elaborate the precautions to be taken in cascading stages of a network from
the point of view of noise reduction?
(b) What is the need for band limiting the signal towards the direction of increas-
ing SVR? [8+8]
? ? ? ? ?







2 of 2Code No: 07A3EC10 Set No. 4
II B.Tech I Semester Regular Examinations, November 2008
PROBABILITY THEORY AND STOCHASTIC PROCESSES
( Common to Electronics & Communication Engineering, Electronics &
Telematics and Electronics & Computer Engineering)
Time: three hours Max Marks: 80
ans any 5 ques.
All ques. carry equal marks
? ? ? ? ?
1. (a) discuss the terms Joint probability and Conditional probability.
(b) Show that Conditional probability satis?es the 3 axioms of probability.
(c) 2 cards are drawn from a 52-card deck (the ?rst is not replaced):
i. provided the ?rst card is a queen. What is the probability that the second
is also a queen?
ii. Repeat part (i) for the ?rst card a queen and 2nd card a 7.
iii. What is the probability that both cards will be the queen? [4+6+6]
2. (a) De?ne cumulative probability distribution function. explain distribution func-
tion speci?c properties.
(b) The random variable X has the discrete variable in the set {-1,-0.5, 0.7, 1.5, 3}
the corresponding probabilities are presumed to be {0.1, 0.2, 0.1, 0.4, 0.2}. Plot
its distribution function and state is it a discrete or continuous distribution
function. [8+8]
3. (a) A random variable X is uniformly distributed in the interval (-5 , 15). a different
random variable Y = e-x/5
is formed. obtain E[Y] and fY (y).
(b) discuss the subsequent terms:
i. Conditional Expected value
ii. Covariance.
(c) obtain the Expected value of the number on a die when thrown. [8+6+2]
4. (a) State and prove central limit theorem.
(b) obtain the density of W = X + Y, where the densities of X and Y are presumed
to be:
fX(x) = [u(x) - u(x - 1)], fY (y) = [u(y) - u(y - 1)] [8+8]
5. (a) Write the expression for expected value of a function of random variables and
prove that the mean value of a weighted sum of random variables equals the
weighted sum of mean values.
(b) X is a random variable with mean ¯ X = 3, variance s2
X = 2.
i. Determine the 2nd moment of X about origin
ii. Determine the mean of random variable y = where y = -6X +22. [8+8]
1 of 2Code No: 07A3EC10 Set No. 4
6. (a) Prove that autocorrelation function of a random process is even function of t .
(b) Prove that RXX (t ) = RXX (0). [8+8]
7. (a) A WSS noise process N(t) has an autocorrelation function RNN (t ) = Pe-3|t|
where p is a constant. obtain and sketch its power spectrum.
(b) Consider the ?gure shown in ?gure 7.
where X(t), Y(t) are random processes & X(t) is WSS. obtain the relation
ranging from SY Y (?) and SXX(?). [8+8]
Figure 7
8. (a) A Signal x(t) = u(t) exp (-at ) is applied to a network having an impulse
response h(t)= ? u(t) exp (-? t). Here a & ? are real positive constants.
obtain the network response?
(b) 2 systems have transfer functions H1( ?) & H2( ?). Show the transfer
function H(?) of the cascade of the 2 is H(?) =H1(?) H2 (?).
(c) For cascade of N systems with transfer functions Hn(?) , n=1,2,........N show
that H(?) = pHn(?). [6+6+4]
? ? ? ? ?
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Earning:   Approval pending.