Random Signal Analysis - May 2015

Electronics & Telecomm. (Semester 5)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.

1 (a) State and prove Baye's theorem. (5 marks)

1 (b) A certain test for a particular cancer is known to be 95% accurate. A person submits to the test and the result are positive. Suppose that the person comes from a population o 100,000 where 2000 people suffer from that disease. What can we conclude about the probability that the person under test has that particular cancer? (5 marks)

1 (c) Let X and Y be independent, uniform r.v.'s in (-1, 1). Compute the pdf of V=(X+Y)². (5 marks)

1 (d) If the spectral density of a WSS process is given by $$ \begin {align*} S(w)&= b (a-|w| )/a &, |w|\le a \\ &=0 &, |w|>a \end{align*} $$ Find the autocorrelaton function of the process. (5 marks)

2 (a) State and prove Chapman-Kolmogorov equation. (10 marks)

2 (b) The joint density function of two continuous r.v.'s X and Y is $$ \begin {align*} f(x,y) & = exy &0< x<4, 1/,y<5 \\ &=0 & otherwise \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*} $$ i) Find the value of constant C.
ii) Find P(X≥3,   Y≤2)
iii) Find marginal distribution function of X. (10 marks)

3 (a) Explain strong law of large number and weak law of large numbers. (5 marks)

3 (b) Explain the central limit theorem. (5 marks)

3 (c) A distribution with unknown mean μ has variance equal to 1.5. Use central limit theorem to find how large a sample should be taken from the distribution in order that the probability will be at least 0.95 that the sample mean will be within 0.5 of the population mean. (10 marks)

4 (a) Given a.r.v. Y with chracteristic function. ϕ(w)=E(e^jwT) and a andom process defined by X(t)= cos (λt+Y), show that X(t) is stationary in wide sense if ϕ(1)=ϕ(2)=0. (10 marks)

4 (b) Define an ergodic process. Determine whether the stochastic process X(t)=A sin(t)+Bcos(t) is ergodic. Here A & B are normally distributed independent r.v.'s with zero mean and equal standard deviation. (10 marks)

5 (a) The joint probability function of two discrete r.v.'s X and Y is given by f(x,y)=e(2x+y), where x and y can assume all integers such that 0≤x≤2, 0≤y≤3 and f(x,y)=0 otherwise. Find E(X), E(Y), E(XY), E(X²), E(Y²), var(X), var(Y), cov (X,Y) and ρ. (10 marks)

5 (b) State and explain various properties of autocorrelation function and power spectral density function. (10 marks)

6 (a) "The transition probability matrix of Markov Chain is $$ \ \ 1 \quad \ \ 2 \ \quad \ 3 \\ \begin{matrix}1\\2 \\3 \end{matrix} \begin{bmatrix} 0.3 &0.4 &0.1 \\0.3 &0.4 &0.3 \\0.2 &0.3 &0.5 \end{bmatrix} $$ Find the limiting probabilities." (10 marks)

Write a short notes on any two of the following:

6 (b) (i) Markov chains. (5 marks)

6 (b) (ii) Little's formula. (5 marks)

6 (b) (iii LTI systems with stochastic input. (5 marks)

6 (b) (iv) M/G/1 queuing system. (5 marks)