Jawaharlal Nehru Technological University Kakinada 2007 B.Tech Computer Science and Engineering PROBABILITY THEORY AND STOCHASTIC PROCESS (4) - Question Paper

Code No: R059210401 Set No. 1
II B.Tech I Semester Regular Examinations, November 2007
PROBABILITY THEORY AND STOCHASTIC PROCESS
( 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) explain Joint and conditional probability.
(b) When are 2 events stated to be mutually exclusive? discuss with an example?
(c) Determine the probability of the card being either red or a king when 1 card
is drawn from a regular deck of 52 cards. [6+6+4]
2. (a) describe rayleigh density and distribution function and discuss them with their
plots.
(b) describe and discuss the guassian random variable in brief?
(c) Determine whether the subsequent is a valid distrbution function. F(x) = 1-
exp(-x/2) for x ) 0 and 0 elsewhere. [5+5+6]
3. (a) State and prove properties of characteristic function of a random variable X
(b) Let X be a random variable described by the density function
fX(x) = _ 5
4 (1 - x4) 0 < x _ 1
0 elsewhere
. obtain E[X] ,E[X2] and variance. [8+8]
4. The joint space for 2 random variables X and Y and corresponding probabilities
are shown in table
obtain and Plot
(a) FXY (x, y)
(b) marginal distribution functions of X and Y.
(c) obtain P(0.5 < X < 1.5),
(d) obtain P(X _ 1,Y _ 2) and
(e) obtain P(1 < X _ 2,Y _ 3).
X, Y 1,1 2,2 3,3 4,4
P 0.05 0.35 0.45 0.15
[3+4+3+3+3]
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) 2 random variables X and Y have joint characteristic function
fX, Y(?1,?2) = exp(-2?2
1-8?2
2).
i. Show that X and Y are zero mean random variables.
1 of 2
ii. are X and Y are correlated. [8+8]
6. Let X(t) be a stationary continuous random process that is differentiable. Denote
its time derivative by X? (t).
(a) Show that E h •
× (t)i = 0.
(b) obtain R× ?× (t ) in terms of R×× (t )sss [8+8]
7. (a) Derive the expression for PSD and ACF of band pass white noise and plot
them
(b) describe different kinds of noise and discuss. [8+8]
8. (a) describe the subsequent random processes
i. Band Pass
ii. Band limited
iii. Narrow band. [3×2 = 6]
(b) A Random process X(t) is applied to a network with impulse response h(t) =
u(t) exp (-bt)
where b > 0 is ? constant. The Cross correlation of X(t) with the output Y
(t) is known to have the identical form:
RXY (t ) = u(t )t exp (-bY)
i. obtain the Auto correlation of Y(t)
ii. What is the avg. power in Y(t). [6+4]
? ? ? ? ?

Earning:   Approval pending.