ٹج

b. Obtain the mean and the moment generating function of a Poisson 4
random variable with parametera.

1

a. If X is astandard normal random variable, then find the pdf of 5

b. If Xand Y are independent and identically distributed uniform 4 random
variables in the interval [0, 1], then find the pdf of 2 = max(X, ४).

PART B

a. Determine the expectation vector and covariance matrix of a two 5
dimensional random vector [XI X2] 1 described by the joint probability
density function
(20 5% < ‏ود‎ 1
१५२ (२५, %2) = 10 otherwise.

b. The joint probability density function of two random variables X 4 and Y
is given by os x ऽ 1,0 Sy sx otherwise. Find the covariance of X and Y.

fav Gy) = ۳

a. Explain a Poisson counting process. 6

b. Prove that the interarrival times of a Poisson counting process 3 follows
an exponential distribution.

a. Find the mean and variance of a continuous random variable with 4

moment generating function
3

Mx(s) = —
3-s

b. A WSS random process X(t) with autocorrelation function 5 =26(0)5 the
input to a continuous time [11 system with impulse response 1(t)=e"u(t).
Find the power spectral density and the autocorrelation of the output
random process Y(t).

PART C
a. State and prove Schwarz inequality for random variables. 4
b. A random variable X has E[X]=8, var[X]=9 and an unknown 4 probability

distribution. Find
i) POX-81>61