JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (October - 2020 - R15)
Answer any five questions. All questions carry equal marks.
Main
State and prove Bayes theorem of probability.
Answer
Bayes' theorem states that for a set of mutually exclusive and collectively exhaustive events B1, B2, ..., Bn that partition the sample space, the conditional probability of an event Bi given that event A has occurred is given by:
To prove this, start with the definition of conditional probability for any two events A and Bi:
Rearranging for P(Bi|A) gives:
Since the events Bj form a partition of the sample space, the event A can be expressed as the union of disjoint events (A ∩ Bj). By the law of total probability, P(A) is:
Substituting this expression for P(A) into the conditional probability formula yields the final theorem.
Find the probability of the card being either red or a king when one card is drawn from a regular deck of 52 cards.
Answer
Let A be the event that the card is red, and B be the event that the card is a king. A standard deck has 52 cards.
1. The number of red cards is 26, so P(A) = 26/52.
2. The number of kings is 4, so P(B) = 4/52.
3. The intersection (A ∩ B) represents red kings. There are 2 red kings (King of Hearts and King of Diamonds), so P(A ∩ B) = 2/52.
Using the addition rule for probabilities:
Simplifying the fraction, the probability is 7/13 or approximately 0.538.
Given P(A) = 1/3, P(B) = 1/2, P(A∩B) = 1/5, then find P(AUB) and P(Ac ∩ Bc).
Answer
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Differentiate between the joint probability and conditional probability.
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The mean of Binomial distribution is 3 and the variance is 9/4. Find i) The value of n ii) P(x > 7).
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Describe the properties of moment generating function.
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Mention about the characteristics of the Gaussian and Rayleigh density Functions and with their distributions graphs in detail.
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Mention about the Statistical Independence and Show that the variance of a weighted sum of uncorrelated random variables equal the weighted sum of the Variances of the random Variables.
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Discuss the Joint Central Moments.
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If X and Y are two random variables which are Gaussian. If a random Variable Z is defined as Z=X+Y, Find fZ(Z).
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Distinguish between the Ergodic and Mean-Ergodic Processes.
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Show that PSD and auto correlation function of random process form a Fourier transform pair.
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A WSS random process x(t) is applied to the input of an LTI system whose impulse response is 5t e-2t. The mean of x(t) is 3. Find the output of the system.
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