Random Signal Analysis - Dec 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) Explain concept of power spectral density.(5 marks) 1 (b) State and prove Central Limit Theorem.(5 marks) 1 (c) Explain properties of cross correlation function.(5 marks) 1 (d) State and prove Baye's theorem.(5 marks) 2 (a) Box 1 contains 5 white balls and 6 black balls. Box 2 contains 6 white & 4 black balls. A box is selected at random and then a ball is chosen at random from the selected Box
i) What is the probability that the ball chosen will be a white ball
ii) Given that the ball chosen is white what is the probability that came from box1.(10 marks) 2 (b) Give the properties of CDF, PDF and PMF.(10 marks) 3 (a) Explain concept of conditional probability and properties of conditional probability.(10 marks) 3 (b) Explain what do you mean by?
i) Deterministic system
ii) Stochastic system
iii) Memoryless system(3 marks) 3 (c) Prove that if input to memoryless system is strict sense stationary (SSS) process then output is also strict sense stationary.(7 marks) 4 (a) Explain Random process, define ensemble mean, Auto correlation and Auto covariance of the process in terms of indexed random variables in usual mathematical forms.(10 marks) 4 (b) Let Z=X+Y. Determine pdf of Z f_z (Z).(10 marks) 5 (a) State and prove Chapman Kolmogorov equation.(10 marks) 5 (b) Explain Chebyshev's Inequality with suitable example.(10 marks) 6 (a) The joint probability density function of two random variables is given by $$ F_{xy}(x, y)=15 \ e^{-3x-3y}; \ \ x\ge 0, y\ge 0 $$ i) Find the probability that x<2 and y>0.2
ii) Find the marginal densities of X and Y
iii) Are X and Y Independent?
iv) Find E(x/y) and E(y/x).(10 marks) 6 (b) Write short notes on following special distributions
i) Poisson distributions
ii) Rayleigh distributions
iii) Gaussian distributions(10 marks)