Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (September - 2021 - R13)
Answer any five questions. All questions carry equal marks.
Main
A binary communication system transmits a 0 and 1 with equal probabilities. Due to the noise in the channel, a transmitted 1 is received as a 0 with a probability of 0.25 and a transmitted 0 is received as 1 with a probability of 0.125. (i) If a message received, find the probability for it to be a 1. (ii) If a zero is received, find the probability that the transmitted is a 0.
Answer
Let T0 and T1 be the events that 0 and 1 are transmitted, and R0 and R1 be the events that 0 and 1 are received. Given P(T0) = P(T1) = 0.5. The conditional probabilities are P(R0|T1) = 0.25, P(R1|T1) = 0.75, P(R1|T0) = 0.125, and P(R0|T0) = 0.875.
(i) Using the Law of Total Probability, the probability of receiving a 1 is:
(ii) Using Bayes' Theorem, the probability that a 0 was transmitted given that a 0 was received is:
What is the probability that a 6 is obtained on one of the dice in a throw of two dice, if the sum of the two faces is 7.
Answer
Let A be the event that a 6 is obtained on at least one die, and B be the event that the sum of the two faces is 7. The total outcomes in the sample space for two dice is 36. The outcomes for event B are {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, so n(B) = 6. The outcomes in B that also contain a 6 (event A ∩ B) are {(1,6), (6,1)}, so n(A ∩ B) = 2. The conditional probability is:
Find the probability of getting sum of the two faces equal to four, atleast once in three tosses of a pair of fair dice.
Answer
Sign in to view the complete answer paper.
A box X contains 3 white and 2 black balls. Box Y contains 2 white and 4 black balls. If one bag is selected at random and a ball is selected from it, find the probability that the ball is black.
Answer
Sign in to view the complete answer paper.
A binary communication source X transmits a 0 and 1, such that P(X=0)=1/7. Due to the noise in the channel, a transmitted 1 is received as a 0 with a probability of 3/8 and a transmitted 0 is received as a 1 with a probability of 1/4. Find the average probability of error in the communication system.
Answer
Sign in to view the complete answer paper.
The probability function of a random variable X is given by f(x) = (2/81)x for -3 < x < 6, and equal to zero otherwise. Find the density of the random variable Y = (1/12)(3 - X).
Answer
Sign in to view the complete answer paper.
Find the Moment generating function of a random variable, uniform over (a, b) and find its first two moments about origin, from its Moment Generating function.
Answer
Sign in to view the complete answer paper.
Find the Coefficient of skewness of a random variable, uniform over (4,6).
Answer
Sign in to view the complete answer paper.
What is difference between first order stationary, second order stationary and wide sense stationary.
Answer
Sign in to view the complete answer paper.
The joint density function of two continuous random variables X and Y is given as f(x,y)=2 for 0 < x < 1, 0 < y < x; and is zero elsewhere. Find the marginal densities of X and Y.
Answer
Sign in to view the complete answer paper.
X and Y are two independent random variables with the individual densities f(x) = 0.25δ(x+1) + 0.5δ(x) + 0.25δ(x-1) and f(y) = 0.5δ(y+1) + 0.5δ(y-2). Find the density of their sum.
Answer
Sign in to view the complete answer paper.
The joint probability matrix of two random variables X and Y is given as P(x,y) = [[0, 1/8, 1/16], [1/16, 1/16, 0], [1/8, 0, 1/8], [3/16, 1/16, 1/4]]. Find the conditional distribution of X, given Y=1. Check X and Y for independence.
Answer
Sign in to view the complete answer paper.
If X and Y are two independent random variables with uniform density over (-1,1) and (-2,1) respectively, find the P(X + Y ≤ -2).
Answer
Sign in to view the complete answer paper.
A random Process is defined as X(t)=3Cos(2πt+Y), where Y is a random Variable with p(Y=0)=p(Y=π)=1/2. Find the mean and Variance of the Random Variable X(2).
Answer
Sign in to view the complete answer paper.
If X(t) and Y(t) are two independent Stationary Processes, find the Auto correlation function of the process Z(t)=2X(t)Y(t).
Answer
Sign in to view the complete answer paper.
A stationary random process X(t) has spectral density Sxx(ω)=25/(ω^2+25) and an independent stationary process Y(t) has the spectral density Syy(ω)=ω^2/(ω^2+25). If X(t) and Y(t) are of zero mean, find the (a) PSD of Z(t)=X(t)+Y(t) (b) cross spectral density of X(t) and Z(t). State and prove the results used.
Answer
Sign in to view the complete answer paper.