What are the elementary properties of probability Consider an experiment of drawing two cards at random from a bag containing four cards marked with the integers 1 through 4.
Find the sample space of the experiment if the first card is replaced before the second is drawn.

(ii) Find the sample space of the experiment if the first card is not replaced.
State the Baye's theorem of conditional probability. A lot contains 100 semiconductor chips, out of these 20 are defective. Two chips are selected randomly, without replacement, from the lot. What is the probability that the first one selected is defective?

(ii) What is the probability that the second one selected is defective given that the first one was defective

(iii) What is the probability that both are defective? Define discrete random variable and probability mass function. What is random process, describe it in detail Define and explain Markov process. Define the power spectral density. What are the various properties of it List out four types of estimations. Explain the queuing system.

2. Consider the switching networks shown in figure Let A1, A2 and A3, denote the events that the switches s1, s2 and s3 are closed, respectively. Aab denotes the event that there is a closed path between terminals a and b. Express Aab in terms of Al, A2 and A3 for each of the networks shown.

<img src='./qimages/15132-2.jpg'>

3. Define probability density function. Let X be a continuous random variable X with pdf

fx(X) {kx
0 otherwise

where k is a constant.
Determine the value of k and sketch Find and sketch the corresponding cdf Find x 2).

4. Suppose the joint pmf of a bivariate r.v. is given by:

Pxy (xi yi)
0 otherwise
Are X and Y independent Are X and Y uncorrelated

5.(a) Differentiate between Markov chain and Markov process.
Derive a two state Markov process and how it is used in digital communications?

6. Consider a Markov chain with two states and transition probability matrix.

P 1/4
1/2 Find the stationary distribution p of the chain. Find Lim n->infinite pn. Find Lim n->infinite pn by first evaluating PH.

7. Let and be defined by
X(t) U cos(wot) V sin(wot) cos(wot) sin(wot)
where wo, is constant and U and V are independent random variables both having zero mean and variance O^2. Find the cross-correlation function of and Find the cross power spectral density of and Y(t).

8. State and explain the following: Discrete random variable. Probability density function.

(c) Cumulative distribution function.