JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (February - 2024 - R15)
i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define the distribution function of a discrete random variable X.
Answer
The distribution function, also known as the cumulative distribution function (CDF), of a discrete random variable X is defined as the probability that the random variable X takes on a value less than or equal to a specific real number x. It is denoted by FX(x) and is given by the formula:
where the summation is performed over all possible values xi of the random variable X that are less than or equal to x.
Describe terms joint probability and conditional probability with an example.
Answer
Joint probability refers to the probability of two or more events occurring simultaneously. For two events A and B, it is denoted as P(A ∩ B) or P(A, B). For example, in a single toss of two fair coins, the joint probability of getting a Head on the first coin and a Head on the second coin is P(H1 ∩ H2) = 1/4.
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B) and defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. For example, if a die is rolled and we know the result is an even number (B = {2, 4, 6}), the conditional probability that the result is a 2 (A = {2}) is P(A|B) = (1/6) / (3/6) = 1/3.
When N random variables are said to be jointly Gaussian?
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Determine whether the given function is a valid probability distribution function or not? Write the properties used.
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Define the joint moments about the origin.
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The joint density function of two discrete random variables X and Y is: Find the value of the constant k.
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What is the difference between random sequence and random process?
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Verify
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Give any two spectral characteristics of the system response.
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Verify that the cross spectral density of two uncorrelated stationary random processes is an impulse function.
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Part B
Explain the concepts: (i) Random variables and its types (ii) Axioms of probability.
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Determine the probability of the card being either red or a king when one card is drawn from a regular deck of 52 cards.
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State and prove Bayes theorem.
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Distinguish between mutually exclusive events and independent events. Consider the experiment of rolling of two fair dice simultaneously and represent its sample space. Also give examples of terms mentioned above related to this experiment.
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Find the Moment generating function of a uniform random variable distribute over (A,B) and find its first and second moments about origin, from the Moment generating function.
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Prove the following:
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Derive the Poisson density function and find mean and variance.
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Write about Chebychev’s inequality and mention about its characteristic function.
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Two statistically independent random variables X and Y have mean values E[X] = 2 and E[Y] = 4. They have second moments E[X^2] = 8 and E[Y^2] = 25. Find Variance of W = 3X-Y.
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Discuss about transformation of multiple random variables.
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Define joint distribution function and write its properties.
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The joint characteristic function of two random variables is given by
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List and explain the properties of Covariance.
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Explain in detail about Poisson random process.
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Describe the condition for wide sense stationary random process.
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Find the auto correlation function for white noise, shown in the figure below.
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The autocorrelation function of a random process
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Derive the relation between PSDs of input and output random process of an LTI system.
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Prove that PSD and ACF of random process form a Fourier transform pair.
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Discuss the transmission of random process through linear system.
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