Random Signal Analysis - May 2013

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 the Chebychev?s inequality and explain.(5 marks) 1 (b) What do you mean by steady state distribution of Markov chain.(5 marks) 1 (c) Suppose X and Y are two random variables when do you say that X and Y are
a) Orthogonal
b) Uncorrelated (5 marks) 1 (d) What is the difference between a Random variable and a Random process? (5 marks) 1 (e) State and explain Baye's theorem. (5 marks) 2 (a) A certain test for a particular cancer is known to be 95% accurate. A person submits to the test and the results are positive. Suppose that the person comes from a population of 1,00,000(one lakh) where 2000 people suffer from that disease , what can we conclude about the probability that the person under test has that particular cancer? (10 marks) 2 (b) Explain with suitable examples Continuous, Discrete and Mixed type random variable. (10 marks) 3 (a) Explain the concept of conditional probability and the properties of conditional probability.(10 marks) 3 (b) Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white and 5 blue balls. If we let X and Y denote respectively the number of red and white balls chosen.
Find :-
(i) The joint probability distribution of (X,Y)
ii) Probability mass function of X
(iii) Probability mass function of Y
(10 marks) 4 (a) $$ Suppose \ f_X(X)=\dfrac {2X}{\pi^2}, 0<X< \pi \ and \ y=\sin x \ Determine \ f_y(y) $$(10 marks) 4 (b) Compare PDF of Binomials and Poison Random variable. A spacecraft has 1,00,000 components. The probability of any one component being defective is 2×10^-5. The mission will be in danger if five or more components become defective. Find a probability of such an event. (10 marks) 5 (a) Define Central limit theorem and give its significance (5 marks) 5 (b) Describe the sequence of random variables. (5 marks) 5 (c) State and prove Chapman-Kolmogorov equation. (10 marks) 6 (a) Explain what do you mean by?
(i)Deterministic system
(ii) Stochastic system
(iii) Memory-less system.
Prove that if input to memory-less system is strict sense stationary(SSS) process x(t), the output y(t) is also SSS. (10 marks) 6 (b) If a random process is given by x(t)=100cos(100t+ θ) where θ is uniformly distributed over (-π,π) , prove that {x(t)} is correlation ergodic. (10 marks) 7 (a) Explain power spectral density function. State its important properties and prove any one of the property. (10 marks) 7 (b) Consider a random process x(t) that assumes the values=±1. Suppose that x(t)= ±1 with probability 1/2 and suppose that x(t) then changes polarity with each occurrence of an event in a poison process of rate α. Find the mean, variance and Auto-Covariance of x(t). (10 marks)