Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (September/October - 2023 - R15)
Note: 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 joint probability.
Answer
The joint probability of two events A and B, denoted as P(A ∩ B), is the probability that both events occur simultaneously in a single trial of an experiment.
Define probability based on axioms.
Answer
Probability is defined by Kolmogorov's axioms: 1) The probability of any event A is non-negative, P(A) ≥ 0. 2) The probability of the sample space S is unity, P(S) = 1. 3) For any sequence of mutually exclusive events A1, A2, ..., the probability of their union is the sum of their individual probabilities, P(∪ Ai) = Σ P(Ai).
What is the significance of standard deviation of a random variable?
Answer
Sign in to view the complete answer paper.
Explain Chebyshev’s Inequality.
Answer
Sign in to view the complete answer paper.
Express covariance in terms of correlation and mean values.
Answer
Sign in to view the complete answer paper.
What is joint characteristic function of random variables X and Y?
Answer
Sign in to view the complete answer paper.
Define mean-ergodic process.
Answer
Sign in to view the complete answer paper.
Differentiate time average and statistical averages.
Answer
Sign in to view the complete answer paper.
Is it possible to represent the random signals in frequency domain by using Fourier transform? Justify your answer.
Answer
Sign in to view the complete answer paper.
What is RMS bandwidth of a power spectrum?
Answer
Sign in to view the complete answer paper.
Part B
State and prove Bayes theorem.
Answer
Sign in to view the complete answer paper.
A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are 0.03 and 0.02 respectively. It is also known that B is more likely to fail (probability 0.05) if A has failed. a) What is the probability of accidentally missile launch? b) What is the probability that A will fail if B has failed? c) Are the events “A fails” and “B fails” statistically independent?
Answer
Sign in to view the complete answer paper.
Derive the expression for density function of a transformed random variable, if the transformation is monotonic and the random variable is continuous.
Answer
Sign in to view the complete answer paper.
The characteristic function of a random variable X is given as
Answer
Sign in to view the complete answer paper.
Write and explain the properties of joint density function of random variables X and Y.
Answer
Sign in to view the complete answer paper.
Statistically independent random variables X and Y have respective densities
Answer
Sign in to view the complete answer paper.
Define cross – correlation function. State and prove the properties of cross-correlation function.
Answer
Sign in to view the complete answer paper.
A stationary random process X(t) is applied to the input of a system for which
Answer
Sign in to view the complete answer paper.
Derive the relationship between power spectrum and autocorrelation function. Also list the properties of power spectrum.
Answer
Sign in to view the complete answer paper.
Explain the power spectrum of response, cross power density spectrum of input and output of the linear system with random inputs.
Answer
Sign in to view the complete answer paper.