# Random Variables and Stochastic Processes July-2011

UNIT – I

1 (a) Show that for any events A and B in S P(B)=P (B/A) P( A)+p(b/X) P( A)

(b) Let A and B be events in a sample space S. Show that if A and B are independent, then show that :

(i)A and Bare independent

(ii)A and B are independent

(iii)Xan(i B are independent.

OR

1  (a) In a system, there are n-components connected in series. This system works successfully. The operation of each component , is independent to each other. -The probability of successful operation of the components is p ;z = 1,2,3, n- Find the probability that the system functions successfully.

The corresponding diagram is shown in Fig.

(b) Let the telegraph source is generating two symbols, dots and dashes. It is observed that the dots were twice as likely to occur as dashes. Determine the probabilities of the dot’s occurring and the dash’s occurring.

UNIT – II

2. (a)  What is random variable ? Explain all the types of random variables with suitable example.

(b) Show that the pdf of normal random variable X satisfies the

OR

(a) Give the applications of the Poisson’s random variables.

(b)   Prove the reproductive property of independent Poisson’s random variable. Hence find the probability of 5 or more telephone calls arriving in a 9 min. period in a college switch – board, if the telephone calls that are received at the rate of 2 every 3 min. follows a Poisson distribution.

UNIT – III

3. (a)  If A – |X < xj and B = {F < >’} are statistical independent events, then, what shall be the effects on the following :

(i) Joint probability distribution function Fxy(x,y)

(ii) Joint probability density function fxy(x,y)

(iii) Conditional distribution

(iv) Conditional density.

(b) The    joint pdf of a bivariate RY (X, Y) is given as where K is a constant

(i)   determine K

(ii) are X and Y independent ?

OR

(a)  Explain all the properties of joint pdf and joint cdf

(b) If the joint pdf of (X, Y) is given by fXY(x,y) = x +y;0< x, y <1, find the pdf of

UNIT – IV

4. (a)  State and prove the “Chebyshev inequality”

(b) Consider X and Y are two random variables where they are defined as

X = cosy/ and Y – sin y where y/ is another random variable uniformaly distributed over (0,2;r). Show that X and Y are uncorrelated.

OR

(a)  State and prove all the properties of moment generating function.

(b) Consider a random variable X which is uniformly distributed

over (-V4,V4j. Calculate then compare it with the upper bound obtained by chebyshev inequality.

UNIT – V

5. (a)  A stationary random process has an autocorrelation t| 10\1—LL ; III <0.05 function of R^T^~\ 1 °-05 ; elsewhere.

(b) Determine the mean and variance of the process X(t). The psd of N(t) is defined in Fig. 5(b), find out the :

(i)   autocorrelation function of N(t).

(ii)  the average power of N(t).

OR

5 (a), X(t) be a input voltage to the circuit and Y(t) be the output voltage. The process X(t) is a stationary random process with zero mean and its autocorrelation

function is 7(,„,(r) = Determine :

(i)E[Y(t)]

(ii) Sn.(w)

(iii) Ryy(t)

(b) If Y(t) be the output of an LTI system with impulse response h(t) when a WSS random-process X(t) is applied as input. Show that :

(i)  SifY(w) = H(w)Sxx(w)

(iii) Srr(w) = H*(w)Sxy(w)