JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (March - 2021 - R15/R13)
Answer any five questions. All questions carry equal marks.
Main
Define and explain the concepts of Joint and Conditional probability.
Answer
Joint probability of two events A and B, denoted as P(A ∩ B), is the probability that both events occur simultaneously. For any two events in a sample space, it represents the intersection of the sets representing those events.
Conditional probability, denoted as P(A|B), is the probability that event A occurs given that event B has already occurred. It is defined by the formula:
This measure updates the likelihood of an event based on the knowledge of another event's outcome.
How do you explain statistically independent events using Baye’s rule?
Answer
Two events A and B are statistically independent if the occurrence of one does not affect the probability of the other. Mathematically, this implies P(A|B) = P(A) and P(B|A) = P(B).
Using Baye's rule:
Substituting the independence condition P(B|A) = P(B) into the formula yields:
This confirms that for independent events, the conditional probability simplifies to the marginal probability, demonstrating that the information provided by event B is irrelevant to the occurrence of event A.
A bag contains four balls. Two balls are drawn and are found to be white. Find the probability that all the balls are white.
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Define conditional distribution and density function of two random variables X and Y.
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State and prove any three properties of moment generating function.
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Define and explain conditional probability mass function. Give its properties.
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The joint distribution of X and Y is given by
Show that X and Y are independent random variables.
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X(t) is a stationary random process with a mean of 3 and an auto correlation function of 6+5 exp(-0.2 |τ|). Find the second central Moment of the random variable Y = Z-W, where Z and W are the samples of the random process at t = 4 sec and t = 8 sec respectively.
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Find the cross correlation between the processes X(t) = Acos(ωt) + Bsin(ωt) and Y(t) = Bcos(ωt) - Asin(ωt), Where A and B are two standardized Gaussian random variables.
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A stationery random process X(t) has spectral density SXX(ω) = 25/(ω2+25) and an independent stationery process Y(t) has the spectral density SYY(ω) = ω2/(ω2+25). If X(t) and Y(t) are zero mean, find the:
PSD of Z(t) = X(t) + Y(t)
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Cross spectral density of X(t) and Z(t). State and prove the results used.
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Distinguish between ensemble average and time average of a random process.
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A random process is defined as X(t) = A sin(ωt + θ), where A is a constant and θ is a random variable uniformly distributed over (-π, π). Check X(t) for stationarity.
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For two jointly stationary random processes, the cross-correlation function is RXY(τ) = 2e-2τ u(τ). Find the two cross-spectral density functions.
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List the properties of cross power spectral density function.
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Show that the autocorrelation function of a stationary random process is an even function of τ.
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Discuss the properties of conditional distribution function.
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