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## 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)