Sums of independent random variables.
a) Show that for any random variables X, Y , and Z,
E(X + Y + Z) = E(X) + E(Y ) + E(Z) .
b) Show that for independent random variables X, Y , and Z,
Var(X + Y + Z) = Var(X) + Var(Y ) + Var(Z) .
c) Generalize these results to sums of n random variables.
d) Do we need full independence for the variance of a sum to be the sum of the variances, or is pairwise independence sufficient?