Suppose that Y is a discrete random variable with mean and variance ^2 and let U=Y/10.

a. Do you expect the mean of U to be larger than, smaller than, or equal to =E(Y)? Why?

b. Use Theorem 3.4 to express E(U)=E(Y/10) 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 U 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(U)=E{[UE(U)]^2}=E[.01(Y)^2]=.01^2; that is, U=Y/10 has variance .01 times that of Y.

Theorem 3.4: Let Y be a discrete random variable with the probability function p(y). Then the expected value of Y, E(Y), is defined to be: E(Y)= y p(y).

y

Definition 3.5: If Y is a random variable with mean E(Y)=, the variance of a random variable Y is defined to be the expected value of(Y)^2. That is, V(Y)=E[(Y)^2]. The standard deviation of Y is the positive square root of V(Y).