JNTU B.Tech III PROBABILITY AND STATISTICS Previous Question Paper (November/December - 2017 - R15)

Define mathematical expectation. What is the mathematical expectation when a fair die is rolled.

Given the probability function for X (0, 1, 2, 3) with p(X) (0.1, 0.3, 0.5, 0.1). Let Y = X² + 2X, then find (i) the probability function of Y. (ii) The mean of Y.

What is meant by coefficient of correlation? What are its limits?

Prove that the regression lines of Y on X and X on Y intersect at the point (X, Y).

Define the terms (i) critical region and (ii) type-I and Type-II error.

Explain Chi-square distribution. What are its applications?

Explain the terms (i) traffic intensity (ii) transient state in queueing theory.

What is pure birth-death process?

Define Markov process. Give example.

If the transition probability matrix of a Markov chain is [[0, 1], [1/4, 3/4]], find the steady state distribution.

Is the function defined as f(x) = e^-x for x ≥ 0 and 0 for x < 0 a valid density function? If so, determine the probability that variate having this density will fall in the interval (1, 2). Find the cumulative probability function.

Define moment generating function (MGF). Find it for the Poisson distribution. Using MGF, show that the sum of two independent Poisson variates is a Poisson variate.

If the probability of a certain column will fail under a given axial load is 0.045, what is the probability that among 16 such columns (i) at most 1 will fail and (ii) at least three will fail.

Explain normal distribution. In a test on 2000 electric bulbs, it was found that the life of a particular make was normally distributed with an average life of 2040 hours and standard deviation of 40 hours. Estimate the number of bulbs likely to burn for (i) more than 2140 hours and (ii) between 1920 and 2080 hours.

Compute the rank correlation coefficient for the given data.

The following results were obtained in the analysis of data on the yield of dry bark in ounces (Y) and age in years (X) of 200 cinchona plants: Average: X=9.2, Y=16.5; SD: X=2.1, Y=4.2; Correlation: +0.84. Construct the two lines of regression and estimate the yield of dry bark of a plant of age 8 years.

The relation between X and Y is given by f(x,y) = (x+y)/21 for 0 ≤ x ≤ 2, 0 ≤ y ≤ 3. (i) Test whether f(x, y) is a valid pdf. (ii) Find marginal density functions and check whether X and Y are independent.

The lines of regression in a bi-variate distribution are X + 9Y = 7 and Y + 4X = 49/3. Find (i) mean of X and Y. (ii) Coefficient of correlation.

A random sample of size 100 is taken from an infinite population having the mean μ = 76 and the variance σ² = 256. What is the probability that x-bar will be between 75 and 80?

Two independent samples had the following values. Sample I: 5 6 4 9 12 15 13 11; Sample II: 9 11 10 13 10 6 9. Is the difference between the means of the sample significant? Test at 5% LOS.

A die is thrown 256 times. An even digit turns up 156 times. Can we say that the die is unbiased.

The following is the distribution of the number of trucks arriving at a company’s warehouse. Fit a Poisson distribution and test for goodness of fit at the level of significance 0.05.

In an M/M/1 queueing system, show that the average number of customers is L = ρ/(1-ρ) = λ/(μ-λ).

Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity is limited to 4 trains, and assuming Poisson arrivals and departures, determine the probability that the yard is empty. Also determine the average number of trains in the system.

Explain pure birth-death processes. Explain the role of exponential distribution in modeling such processes.

At a certain petrol pump, customers arrive in a Poisson process with an average time of five minutes between arrivals. The time intervals between services at the petrol pump follow exponential distribution and the mean time taken to service a unit is two minutes. Find (i) the average time a customer has to wait. (ii) By how much time the flow of customers be increased to justify the opening of another service point, where the customer has to wait for five minutes for the service.

Define stochastic process and stochastic matrix. Give examples. When is a stochastic matrix said to be regular?

Find the nature of states of the Markov chain with transition probability matrix P = [[0, 1, 0], [1/2, 0, 1/2], [0, 1, 0]].

Define transition probability and explain Chapman-Kolmogorov equations.

A fair die is tossed repeatedly. If X_n denotes the maximum of the numbers occurring in the first n tosses, find the transition probability matrix of the Markov chain, and classify its states.