Jawaharlal Nehru Technological University Kakinada 2007 B.Tech Computer Science and Systems Engineering PROBABILITY THEORY AND STOCHASTIC PROCESS (1) - Question Paper

Code No: R059210401 Set No. 4
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
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1. (a) describe and discuss the subsequent with an example:
i. Equally likely events
ii. Exhaustive events
iii. Mutually exclusive events
(b) provide the classical definition of probability.
(c) obtain the probability of 3 half-rupee coins falling all heads up when tossed
simultaneously. [6+4+6]
2. (a) What is poisson random variable? discuss in brief.
(b) What is binomial density and distrbution function?
(c) presume automobile arrives at a gasoline station are poisson and occur at an
avg. rate of 50/hr. The station has only 1 gasoline pump. If all cars are
presumed to require 1 minute to find fuel. What is the probability that a
waiting line will occur at the pump? [5+5+6]
3. (a) describe moment generating function.
(b) State properties of moment generating function.
(c) obtain the moment generating function about origin of the Poisson distribution.
[3+4+9]
4. provided the function f(x, y) = _ (x2 + y2)/8p x2 + y2 < b
0 elsewhere
(a) obtain the constant ‘b’ so that this is a valid joint density function.
(b) obtain P(0.5b < X2 + Y2 < 0.8b). [7+9]
5. 3 statistically independent random variables X1,X2 and X3 have mean values
¯X
1= 3, ¯X2= six and ¯X3= -2. obtain the mean values of the subsequent functions.
(a) g(X1,X2,X3) = X1 + 3X2 + 4X3
(b) g(X1,X2,X3) = X1 X2 X3
(c) g(X1,X2,X3) = -2X1,X2 -3X1 X3 + 4X2 X3
(d) g (X1,X2,X3) = X1+X2+X3. [16]
1 of 2
6. Statistically independent zero mean random processes X(t) and Y(t) have auto
correlations functions
RXY (t ) = e - |_| and
RYY(t ) = cos (2_t ) respectively.
(a) obtain the auto correlation function of the sum W1(t) = X(t) + Y(t)
(b) obtain the auto correlation function of difference W2(t) = X(t) - Y(t)
(c) obtain the cross correlation function of W1(t) and W2(t). [5+5+6]
7. (a) obtain the ACF of the subsequent PSD’s
i. S__(?) = 157+12!2
(16+!2)(9+!2)
ii. S__(?) = 8
(9+!2)2
(b) State and Prove wiener-Khinchin relations. [8+8]
8. A random noise X(t) having power spectrum SXX(?) = 3
49+!2 is applied to a to a
network for which h(t) = u(t)t2 exp(-7t). The network response is denoted by Y(t)
(a) What is the avg. power is X(t)
(b) obtain the power spectrum of Y(t)
(c) obtain avg. power of Y(t). [5+6+5]
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Earning:   Approval pending.