Statistics UPSC PYQ 2017

In a community of (a+b) potential voters, a are for abortion and b(b < a) are against it. Suppose that a vote is taken to determine the will of the majority with regard to legalizing abortion. If n(n < b) random persons of these (a+b) potential voters do not vote, what is the probability that those against abortion will win?

Suppose n_1 = 11 and n_2 = 12 observations are made on two random vectors X_1 and X_2 which are assumed to have bivariate normal distribution with a common covariance matrix Sigma, but possibly different mean vectors mu_1 and mu_2. The sample mean vectors and pooled covariance matrix are X_bar_1 = [-1, -1]', X_bar_2 = [2, 1]', S_{pooled} = [ [7, -1], [-1, 5] ]. Obtain Mahalanobis sample distance D^2 and Fisher's linear discriminant function. Assign the observation X_0' = (0, 1) to either population pi_1 or pi_2.

X_1 is the time that a customer takes from getting on line at a service desk in a bank to completion of service and X_2 is the time to wait in line before reaching the service desk (X_1 >= X_2). The joint probability density function of (X_1, X_2) is given by f(x_1, x_2) = { e^(-x_1), 0 <= x_2 <= x_1 < infinity; 0, elsewhere }. Let Y_1 = X_1 + X_2 and Y_2 = X_1 - X_2. Find the joint probability density function of Y_1 and Y_2. Are Y_1 and Y_2 independently distributed? Give reason.

Let y_1 = beta_1 - beta_2 + e_1, y_2 = beta_3 + e_2 and y_3 = beta_1 + beta_2 + e_3, where e_1, e_2 and e_3 are iid N(0, sigma^2). Obtain best linear unbiased estimator (BLUE) of (2beta_1 - beta_2). Is this linear parametric function testable?

Ten percent of a certain population suffer from a serious disease. Two independent tests are given to a person suspected of the disease. Each test gives a correct diagnosis 90% of the time. Find the probability that the person really suffers from the disease, given that--(i) both tests are positive;(ii) only one test is positive.

Consider a simple regression model y = beta_0 + beta_1 x + e based on n observations. Show that the regression sum of squares is beta_1^2 sum_{i=1}^n (x_i - x_bar)^2. Suppose the summarized data on 10 observations yield sum(y_i - y_hat_i)^2 = 3.267, sum y_i = 63.3, sum y_i^2 = 423.49, sum x_i = 139, sum x_i^2 = 2239. Estimate beta_1 assuming positive correlation between X and Y. Also obtain coefficient of determination.

Let {X_n} be a sequence of independent and identically distributed random variables with common probability function P[X = -n] = P[X = n] = 3 / (pi^2 n^2), n = 1, 2, ... Check whether {X_n} obeys the (i) strong law of large numbers and (ii) weak law of large numbers.

Explain the method of systematic sampling and give its merits and demerits. How do you estimate the sampling variance of the estimate of the population total? Show how systematic sampling is a particular case of cluster sampling.

Obtain first canonical correlation and its associated canonical variable pair for the following correlation matrix: rho = [ [1, 1/2, 2/3], [1/2, 1, 2/3], [2/3, 2/3, 1] ] ... wait, transcription from scan verbatim.rho has 4x4 matrix visually: rho = [ [1, 1/2, 2/3, 2/3], [1/2, 1, 2/3, 2/3], [2/3, 2/3, 1, 1/2], [2/3, 2/3, 1/2, 1] ]. Let's write exact text from scan.

Let X_1, X_2, ..., X_n be a random sample from a distribution with probability density function f(x) = theta x^(theta-1), 0 < x < 1, theta > 0. Find the maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) of theta.

Derive the expression for the variance of Horvitz-Thompson estimator (HTE) of population total. Point out the difficulty faced in estimating it. Find Yates-Grundy variance estimator.

If the random variable X follows normal distribution N(u, 1), then find the expectation of Y = (1 - Phi(X)) / phi(X), where Phi and phi denote the cumulative distribution function and probability density function respectively of N(0, 1), the standard normal distribution.

Distinguish between a factorial experiment and a number of single-factor experiments. What is meant by confounding in a factorial experiment? Why is confounding preferred even at the cost of loss of information on the confounded effects?

Let X_1, X_2, ..., X_n be a random sample from a normal population with mean mu and variance 25. For a sample size of n = 16, find the uniformly most powerful (UMP) test for testing the simple null hypothesis H_0 : mu = 5 against the composite alternative hypotheses (i) H_1 : mu > 5 and (ii) H_1 : mu < 5.

An investigator selects 10 one-acre plots by simple random sampling and counts the number of trees (y) on each plot. She also has aerial photographs of the plantation from which she can estimate the number of trees (x) on each plot of the entire plantation. Hence, she knows mu_x = 19.7 and since the two counts are approximately proportional through the origin, she uses a ratio estimate to estimate mu_y. The data yield the following: N = 1000 (plantation size), n = 10 (taken by SRS), y_i = The actual count of trees in one-acre plots, i = 1, 2, ..., 10, x_i = The aerial estimate for each plot, y_bar = 22.10, x_bar = 20.80. Estimate the average number of trees per acre and the standard error of the estimate.

Let X_1, X_2, ..., X_n be a random sample from uniform population over (0, theta) and let M = Max(X_1, X_2, ..., X_n). Show that M is a biased but consistent estimator of theta.

Show that in the model y = Xbeta + epsilon subject to Cbeta = 0, the estimator of beta is beta_c = hat(beta) - (X'X)^-1 C' [C(X'X)^-1 C']^-1 C hat(beta) where hat(beta) = (X'X)^-1 X'y. Note here that y is n x 1, X is n x k, beta is k x 1 and C is q x k of rank q <= k.

State and prove Chebyshev's inequality. Hence or otherwise, prove that (i) P[X - mu| >= (d/2)] <= (4/lambda) and (ii) P[X >= 2lambda] <= 1/2, if X is any positive random variable having both mean and variance equal to lambda.

Explain the concept of balanced incomplete block design (BIBD). What are the conditions for existence of a BIBD? If V = {1, 2, 3, 4, 5, 6, 7}, then form a number of blocks, each of order 3, such that each pair of elements in V is contained in exactly one block.

Using Kolmogorov-Smirnov test, determine whether the sample data given below come from an exponential distribution with mean 4. The observations in the sample are: 0.7, 5.5, 5.7, 4.3, 0.8, 1.1, 5.1, 4.9, 2.0, 3.9, 5.3, 4.2, 5.2, 5.8, 6.0 (Given, D_{13, 0.05} = 0.361, D_{14, 0.05} = 0.349, D_{15, 0.05} = 0.338)

Let X_1, X_2, ..., X_n be independently and identically distributed observations from a distribution with probability density function f_theta(x). If Y = g(X_1, X_2, ..., X_n) is an unbiased estimate of theta, then var_theta(Y) >= 1 / (n E_theta [ (d/dtheta log f_theta(X))^2 ]). Use this inequality to show that the sample mean is a uniformly minimum variance unbiased estimator of mean of a Poisson distribution.

Let X_1, X_2, X_3 and X_4 be independent and identically distributed random variables each having probability density function f(x) = { 3(1-x)^2, 0 < x < 1; 0, otherwise }. Find the cumulative distribution function and probability density function of Y = Min(X_1, X_2, X_3, X_4).

Let X = [X_1, X_2, X_3]' have trivariate normal distribution N_3(mu, Sigma), where mu' = [1, 1, 1] and Sigma = [ [1, 1, 1], [1, 2, 2], [1, 2, 3] ]. Show that (X_2 - X_1)^2 + (X_3 - X_2)^2 has a chi-squared distribution. State the degrees of freedom.

Let X be a continuous random variable with the probability density function f(x) = { 2/x^2, if 1 < x < 2; 0, elsewhere }. Find the cumulative distribution function and the probability density function of Y = X^2.

Three sides of an equilateral triangle were measured by 5 persons with the following results: [table with persons A, B, C, D, E and sides a, b, c]. Is there any significant difference between (i) measurements by the persons and (ii) the sides of the triangle?

Y_1, Y_2, Y_3, Y_4 are four independent variables with E(Y_1) = E(Y_3) = theta_1 + theta_3 + theta_4, E(Y_2) = E(Y_4) = theta_1 - theta_2 and V(Y_i) = sigma^2 (for i = 1, 2, 3, 4). Verify whether theta_1 + theta_3 and theta_2 + theta_3 + theta_4 are estimable. If so, obtain their BLUEs. Also obtain the variances of the BLUEs.

For a balanced one-way random effects model Y_{ij} = mu + a_i + e_{ij}, i = 1, 2, ..., a; j = 1, 2, ..., r where a_i ~ N(0, sigma_a^2) vi and e_{ij} ~ N(0, sigma_e^2) vi, j, show that P[F_{a(r-1), a-1} > 1 + nt] where tau = sigma_a^2 / sigma_e^2.

Let x_1, x_2, ..., x_n be n observations, each independently uniformly distributed between theta - 1/2 and theta + 1/2. Find a general form of maximum likelihood estimator (MLE) of theta. Verify that (2Y_1 + 4Y_n - 1) / 6 and (4Y_1 + 2Y_n + 1) / 6 are both MLEs, where Y_1 = Min_i X_i and Y_n = Max_i X_i.