JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (September/October - 2023 - R13)
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
Suppose X has the following probability mass function p(0) = 0.2, p(1) = 0.5, p(2) = 0.3. Calculate
Answer
The expected value E(X2) is calculated as the sum of x2 multiplied by their respective probabilities: E(X2) = (02 * 0.2) + (12 * 0.5) + (22 * 0.3) = 0 + 0.5 + 1.2 = 1.7.
Calculate Var(X) when X represents the outcome when we roll a fair die.
Answer
For a fair die, X takes values {1, 2, 3, 4, 5, 6} each with probability 1/6. The mean E(X) = (1+2+3+4+5+6)/6 = 3.5. The expected value E(X2) = (12+22+32+42+52+62)/6 = 91/6 ≈ 15.167. Thus, Var(X) = E(X2) - [E(X)]2 = 15.167 - (3.5)2 = 15.167 - 12.25 = 2.917.
If X and Y have joint p.d.f f(x, y) = x + y, 0 < x < 1, 0 < y < 1; 0, Otherwise, Check whether X and Y are independent or not.
Answer
Sign in to view the complete answer paper.
Let X and Y have joint density function f(x, y) = 2, 0 < x < y < 1. Find the conditional function of Y given X = x.
Answer
Sign in to view the complete answer paper.
Write any two characteristics of
Answer
Sign in to view the complete answer paper.
Explain the terms sample size and sampling error in a random sampling?
Answer
Sign in to view the complete answer paper.
Explain the notation to represent a Queuing system.
Answer
Sign in to view the complete answer paper.
Define traffic intensity of a Queue.
Answer
Sign in to view the complete answer paper.
Define the State space.
Answer
Sign in to view the complete answer paper.
Give an example of stationary processes and justify your claim?
Answer
Sign in to view the complete answer paper.
Part B
Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50 and variance 25. (i) What can be said about the probability that this week’s production will exceed 75? (ii) What can be said about the probability that this week’s production will be between 40 and 60?
Answer
Sign in to view the complete answer paper.
Let X be a random variable with E(X) = 10 and Var(X) = 25. Find the positive values of ‘a’ and ‘b’ such that Y has expectation 0 and variance 1.
Answer
Sign in to view the complete answer paper.
The number of PCs sold daily at a computer world is uniformly distributed with a minimum of 2000 PC and a maximum of 5000 PCs. Find (i) The probability that daily sales will fall between 2500 and 3000 (ii) What is the probability that the computer world will sell at least 4000 PCs? (iii) What is the probability that the computer world will exactly sell 2500 PCs?
Answer
Sign in to view the complete answer paper.
If a random variable ‘x’ has the M.G.F
Answer
Sign in to view the complete answer paper.
3 balls are drawn at random without replacement from a box containing 2 white, 3 red and 4 black balls. If X denotes the number of white balls drawn and Y denote the number of red balls. Find the P(X,Y).
Answer
Sign in to view the complete answer paper.
The joint p.d.f of random variable X and Y is given by
Answer
Sign in to view the complete answer paper.
Calculate the correlation coefficient for the following heights(in inches) of fathers X and their sons Y. X: 65 66 67 67 68 69 70 72; Y: 67 68 65 68 72 72 69 71
Answer
Sign in to view the complete answer paper.
Let X, Y and Z be uncorrelated random variable with 0 means and standard deviations 5, 12 and 9 respectively. If U = X+Y and V = Y+Z, Find Karl-Pearson’s coefficient of correlation between U and V.
Answer
Sign in to view the complete answer paper.
The mean life of sample of 25 bulbs is found to be 1550 hrs with S.D of 120 hrs. The company manufacturing bulbs claims that the average life of the bulbs is 1600 hrs. Is the claim acceptable at 95% level of significance.
Answer
Sign in to view the complete answer paper.
In a sample of 1,000 people in Maharashtra, 540 are rice eaters and the rest are wheat eaters. Can we assume that both rice and wheat are equally popular in this State at 1% level of significance?
Answer
Sign in to view the complete answer paper.
Two independent samples of sizes 9 and 7 from a normal population at the following values of the variables. Sample 1: 18 13 12 15 12 14 16 14 15; Sample 2: 16 19 13 16 18 13 15. Do the estimates of the population variances differ significantly at 5% level?
Answer
Sign in to view the complete answer paper.
In 120 throws of a single die the following distribution of faces were observed. Faces 1 2 3 4 5 6; Frequency 30 25 18 10 22 15. Can you say that the die is biased?
Answer
Sign in to view the complete answer paper.
Explain (M/M/1) : (∞/FCFS) queueing model.
Answer
Sign in to view the complete answer paper.
A Supermarket has a server servicing at the counter. The customer arrive in a Poisson fashion at the rate of 10 per hour. The service time for each customer is expected with mean 4 minutes. Find: a) The probability of a customer has to wait for the service, b) Average queue length and c) The average time spent by a customer in the queue.
Answer
Sign in to view the complete answer paper.
Explain briefly the main characteristics of Queuing system
Answer
Sign in to view the complete answer paper.
Derive the formula for the probability of having n customers in the system.
Answer
Sign in to view the complete answer paper.
The Probability distribution of the process {X(t)} is given by P[X(t) = n] = (at)^(n-1) / (1+at)^(n+1) for n = 1, 2, 3... and at / (1+at) for n = 0, Show that it is not stationary.
Answer
Sign in to view the complete answer paper.