X and Y take the value 0 with probability 0.25, the value 1 with probability 0.5, and the value 2 with probability 0.25.
Consider independent random variables X and Y with identical probability distributions as given. Let Z = X + X and S = X + Y.
a. Compute the mean and variance of Z directly from its probability distribution.
b. Compute the mean and variance of S directly from its probability distribution.
c. Find E(Z) by using Theorem 7.4 and Var(Z) by using the general addition rule for covariances and the covariance of X with itself (Exercise 13).
d. Find E(Z) by using the fact that Z = 2X and Theorem 7.5. and Var(Z) by using the fact that Z = 2X and Theorem 7.11.
e. Find E(S) and Var(S) from Theorem 7.4 and Theorem 7.9. Why is Var(S) < Var(Z)?