Suppose that Y is a discrete random variable with mean and variance 2 and let W = 2Y . a

Chapter 3, Problem 3.31

(choose chapter or problem)

Suppose that Y is a discrete random variable with mean and variance 2 and let W = 2Y . a Do you expect the mean of W to be larger than, smaller than, or equal to = E(Y )? Why? b Use Theorem 3.4 to express E(W) = E(2Y ) in terms of = E(Y ). Does this result agree with your answer to part (a)? c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of W to be larger than, smaller than, or equal to 2 = V(Y )? Why? d Use Definition 3.5 and the result in part (b) to show that V(W) = E{[W E(W)] 2 } = E[4(Y )2 ] = 42 ; that is, W = 2Y has variance four times that of Y .