a. If X and Y are two continuous random variables,
then show that 4
E[E[Y|X]] = 20).

b. A white Gaussian noise process W(t) with two-sided
power 5
spectral density IS given as input to an
ideal low pass filter with cutoff frequency
B Hz and unity gain. Find the power spectral
density and autocorrelation of the output
random process.

2
PART C

a. State and prove Chebyshev inequality. 6
0. Anormal random variable X has mean value of 5.5 and variance 1. 6
Find an estimate of P(X 2 11) using Chernoff bound.
a. Define the following for a sequence of random variables 4
)( Almost sure convergence
(ii) | Convergence in probability
(iii) | Mean-square convergence
(iv) | Convergence in distribution
b. State and prove Central Limit Theorem for independent and 8
identically distributed random variables.

a. The transition probability matrix P of a two-state Markov chain is 3
given by

1/2 1/2.
Find the steady state distribution of the chain.
b. Derive the Karhunen-Loeve (KL) expansion of a Wiener random
9 process.