JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (April/May - 2018 - R13)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 Marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 Marks.
PART - A
Write the conditions for a function to be a random variable.
Answer
For a function X(s) to be a random variable, it must satisfy two conditions: 1) The set of outcomes {s : X(s) ≤ x} must be an event in the sample space for every real number x, and 2) The probabilities of the events {X = ∞} and {X = -∞} must be zero.
Explain the significance of mathematical model of experiments.
Answer
Mathematical models of experiments allow for the systematic analysis of random phenomena by mapping physical outcomes to numerical values. They provide a rigorous framework to calculate probabilities, predict long-term behavior, and apply statistical tools to real-world systems where uncertainty is inherent, enabling engineers to design robust communication and signal processing systems.
Write short notes on Chebychev’s inequality.
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Define Characteristic function and present generation of moments using it.
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State central limit theorem for the case of equal distributions.
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Write the properties of jointly Gaussian random variables.
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What is a WSS random process?
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Write short notes on Gaussian random process.
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Write the expression for power spectral density.
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Write any three properties of cross-power density spectrum.
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PART - B
A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03, respectively. It is also known that B is more likely to fail (probability 0.06), if A has failed.
What is the probability of an accidental missile launch?
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What is the probability that A will fail, if B has failed?
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Are the events “A fails” and “B fails” statistically independent?
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You (A) and two others (B and C) each toss a fair coin in a two-step gambling game. In step1 the person whose toss is not a match to either of other two is “odd man out”. Only the remaining two whose coins match go on to step2 to resolve the ultimate winner.
What is the probability that you will advance to step2 after the first toss?
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What is the probability you will be out after the first toss?
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What is the probability that no one will be out after the first toss?
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Obtain the moment generating function of a uniformly distributed random variable.
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Obtain the variance of Raleigh random variable.
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A random variable X uniformly distributed in the interval (0, π/2). Consider the transformation Y=sinx, obtain the pdf of Y.
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Obtain the variance of Gaussian random variable.
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The joint characteristic function of two random variables is given by
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X and Y are statistically independent random variables and W = X+Y obtain the pdf of W.
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Write the properties of joint distribution function.
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Prove that the variance of weighted sum of N random variables equals the weighted sum of all their covariances.
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Define autocorrelation function of a random process. Write properties of auto correlation function of a WSS process and prove any three of them.
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A random process X(t) = Acos(ωot) + B sin(ωot) where ωo is a constant and A, B are uncorrelated zero mean random variables with same variances. Check whether X(t) is WSS or not?
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Classify random processes and explain.
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Derive the relationship between cross-power spectrum and cross-correlation function.
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The autocorrelation function of a random process RXX(τ) = 4 cos(ωoτ), where ωo is a constant. Obtain its power spectral density.
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Obtain the average power in the random process X(t) = Acos(ωot+θ) where A, ωo are real constants and θ is a random variable uniformly distributed in the range (0, 2π).
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