December 17, 2011



 Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

1. (a) What are the three axioms to be satisfied by the assigned probabilities? Explain each with an example?
(b) When two dice are thrown determine the probabilities fro axiom 3 for the following three events.
i. A = {sum = 7}
ii. B = {8 ? sum = 11}
iii. C = {10 ? sum} [8+8]

2. (a) suppose that length of an appliance has an exponential distrbution with x=10 years. A used appliance is brought by someone. What is the probabiltiy that it will not fail in the next 5 years.
(b) Suppose that the amount of waiting time of a customer spends at a restaurant has exponential distribution with a mean value of 5 mins. Calculate the probability that a customer will spend more than 10 mins in the restaurant.
(c) Prove that the density function for a discrete random variable is an impulse function [4+4+8]

3. (a) A random variable X has a characteristic function given by
x (?) = 1 - |?| |?| = 1
0 |?| > 1. Find density function
(b) A random variable X has the density function fX(x) = 1
ae-b|x| -8 = x = 8 . Find E[X], E[X2] and variance. [8+8]

4. Given the function f(x, y) = (x2 + y2)/8p x2 + y2 < b
0 elsewhere
(a) Find the constant ‘b’ so that this is a valid joint density function.
(b) Find P(0.5b < x2 + x2 < 0.8b). [7+9]

5. (a) let Y = X1 + X2 + ............+XN be the sum of N statistically independent random variables Xi, i=1,2.............. N. If Xi are identically distributed then find density of Y, fy(y).
(b) Consider random variables Y1 and Y2 related to arbitrary random variables X and Y by the coordinate rotation. Y1=X Cos ? + Y Sin ? Y2 = -X Sin ? + Y Cos ?i. Find the covariance of Y1 and Y2, CY1Y2
ii. For what value of ?, the random variables Y1 and Y2 uncorrelated. [8+8]

6. Sample functions in a discrete random process are constants i.e. X(t) = C = constant. Where C is a discrete random variable having possible values C1=1, C2=2 and C3=3 occurring with probabilities 0.6, 0.3 and 0.1 respectively.
(a) is X(t) deterministic
(b) Find the first order density function if X(t) at any time t. [8+8]

7. (a) Prove the relation between continuous and discrete power spectral densities.
(b) Draw the ACF and PSD of white noise. and hence derive the expression for ACF and PSD of band limited white noise and plot them. [8+8]

8. A random noise X(t) having power spectrum SXX(?) = 3
49+!2 is applied to a to a network for which h(t) = u(t)t2 exp(-7t). The network response is denoted by Y(t)
(a) What is the average power is X(t)
(b) Find the power spectrum of Y(t)
(c) Find average power of Y(t).



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