52032 Pages: 3

PART 13
Answer any two questions

4 a) Obtain the distribution function of a continuous two dimensional random variable 7

(X,Y) with the joint pdf given by

f(x,y) 2 ۳۵ > ‏م > ع‎ ,0O, elsewhere

b) Ifthe joint pdf of (X, Y) is given by 8 f(x,y) ‏بط ع‎ x > 0, y >0,xt+y

0, othe ise

Find P{X SI,1), 2೩೫೪1) { X>2Y }
a) The joint pdfof (X,Y) is 7 f(x,y) 859, 0 > y > x > 10, otherwise
i) Check whether X and Y are independent ii)
FindP( X +
b) Prove that the power spectral density and autocorrelation function of a real WSS 8 process

form a Fourier cosine transform pairhttp://www.ktuonline.com

a) If X(t) P + Qt is the random process where P and Q are independent random 7 variables with
E(P)=p , E(QFq, Var(P)= , Var (ಛಿ) = 02 then find the mean, autocorrelation and autocovariance
of the process

b) Consider the random process X(t)= A cos(ot) + B sin (mt) where A and B are 8 independent
random variables with mean O and equal variance. Show that X(t) is a wss.

PART 0
Answer any two questions

a) Cell phone calls processed by a certain wireless base station arrive according to a 7 Poisson
process with an average of 12 per minute. i) What is the probability that more than two calls
arrive in an interval of length 20 seconds ii) What is the probability that more than 2 calls arrive
in each of two consecutive intervals of length 30 seconds http:::www.ktuonline.com

b) Show that the time between any two consecutive occurrences of a Poisson process 7 is a

random variable following an exponential distribution
c) Find the mean, variance, autocorrelation and autocovariance ofa Poisson process 6

8) Find the positive solution of the equation 2sinx = x using Newton-Raphson method 6

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