Random Signal Analysis - Dec 2011

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) Discuss the properties of cross correlation function(5 marks) 1 (b) If A and B are two independent events then prove that P(A∩B)= P(A) P(B )(5 marks) 1 (c) State and explain Bayes Theorem(5 marks) 1 (d) Suppose five cards to be drawn at random form a standard deck od cards. If all the drwan cards are red what is the probability that all of them are hearts?(5 marks) 2 (a) A random variable has the following exponential probability density function : f(x)=Ke^-|x|. Determine the value of K and corresponding distribution function(10 marks) 2 (b) Define discrete and continuous random variables by giving examples. Discuss the properties of distribution function.(10 marks) 3 The joint probability density function of two random variables is given by
f_x,y(x,y)=15e^-3x-3y : x≥0, y≥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 independant.
(iv) Find E(x/y) and E(y/x)
(20 marks) 4 (a) State and prove the Chapman-Kolmogorov equation.(10 marks) 4 (b) Write short notes on the following special distributions
(i) Poisson distribution (ii) Reyleigh distribution and (iii) Gaussian distribution(10 marks) 5 (a) Suppose X and Y are two random variables. Define covariance and correlation of X and Y. when do we say that X and Y are
(i) Orthogonal
(ii) Independent and
(iii) Uncorrelated> Are uncorrelated variables independent?
(b) What is random process? State four classes of random process giving one example of each.(10 marks) 6 (a) Explain power spectral density function. State its important properties and prove anyone of the property.(10 marks) 6 (b) Prove that if input to LTI system is w.s.s. then the output is also w.s.s.(10 marks) 7 (a) Define Central Limit Theorem and give its significance(5 marks) 7 (b) Describe sequence of ramdom variables(5 marks) 7 (c) A stationary process is givien by X(f)=100 cos[100 t+θ] where θ is a random variable with uniform probability distribution in the interval [-π, π]. Show that it is a wide sense stationary process.(10 marks)