Consider the discrete random variable x with probability distribution p(x). Denote the mean and standard deviation of x as μ and ox, respectively. Now define a new discrete random variable y defined as y = μx - 3x. (a) What is the mean of y, y, in terms of u? (b) What is the standard deviation of y, oy, in terms of ox? (c) Now consider a new random variable z = μx ax, what value of a is the mean of z, z, equal to zero? where a is some constant. For

Let T be a random variable. If E(T) = a +60, find an unbiased estimator of 0. Is the mean- squared error (MSE) of an unbiased estimator always smaller than the MSE of a biased estimator? Give an example to support your answer.

Let X and Y be random variables having the same distribution. Show that Cov(X +Y, X – Y) = 0.

Let X and Y be independent exponential random variables with parameters 2 and 3. (a) What is the probability that X + Y > 1. (b) Conditional on X + Y = 1, what is the distribution of X.