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.