## 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)