## 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)