JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (November/December - 2020 - R16)
Answer any five questions. All questions carry equal marks.
Main
Define the term Independent events. State the conditions for independence of:
i) Any two events A and B.
ii) Any three events A, B and C.
Answer
Two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other. Mathematically, this implies that the joint probability of the events is equal to the product of their individual probabilities.
i) Two events A and B are independent if and only if:
ii) Three events A, B, and C are mutually independent if and only if they satisfy the following four conditions:
- P(A ∩ B) = P(A)P(B)
- P(A ∩ C) = P(A)P(C)
- P(B ∩ C) = P(B)P(C)
- P(A ∩ B ∩ C) = P(A)P(B)P(C)
Define and explain joint and conditional probability.
Answer
Joint Probability refers to the probability of two or more events occurring simultaneously. For two events A and B, the joint probability is denoted as P(A ∩ B) or P(A, B), representing the likelihood that both events happen in the same trial.
Conditional Probability measures the probability of an event occurring given that another event has already occurred. It incorporates new information to update the likelihood of an event. For two events A and B, where P(B) > 0, the conditional probability of A given B is defined as:
This formula shows that the conditional probability is the ratio of the joint probability of both events to the probability of the conditioning event.
Explain the concept of random variable.
Answer
Sign in to view the complete answer paper.
A pair of dice is tossed. Define a random variable X to be the difference of the face values turned up. Determine the probability mass function of X.
Answer
Sign in to view the complete answer paper.
Explain the Rayleigh probability density function.
Answer
Sign in to view the complete answer paper.
Derive the Poisson density function and find mean.
Answer
Sign in to view the complete answer paper.
Find the Moment generating function of exponential distribution.
Answer
Sign in to view the complete answer paper.
Let X be a continuous random variable with PDF fX(x) = 8/x3, x > 2. Find E[W] where W = X/3.
Answer
Sign in to view the complete answer paper.
Define the joint distribution function. Explain how marginal density functions are computed given their joint distribution functions.
Answer
Sign in to view the complete answer paper.
Two independent random variables X and Y have the probability density functions respectively fX(x) = xe-x, x > 0 and fY(y) = 1 for 0 < y < 1, 0 otherwise. Calculate the probability distribution and density functions of the random variable Z = XY.
Answer
Sign in to view the complete answer paper.
State and prove central limit theorem.
Answer
Sign in to view the complete answer paper.
Let X and Y random variables have the joint density function fXY(x,y) = 2, 0 < x < y < 1. Find the marginal density function.
Answer
Sign in to view the complete answer paper.
Explain the classification of random processes with neat sketches.
Answer
Sign in to view the complete answer paper.
The power spectral density of a stationary random process is given by Sxx(ω) = A for -k < ω < k, 0 otherwise. Find the auto correlation function.
Answer
Sign in to view the complete answer paper.
Find the PSD of a random process z(t) = x(t) + y(t) where x(t) and y(t) are zero mean, individual random process.
Answer
Sign in to view the complete answer paper.
Find the Auto correlation and power spectral density function of x(t) = A for 2 < t < 3, 0 otherwise.
Answer
Sign in to view the complete answer paper.