O RANDOM VARIABLES AND DISTRIBUTIONS Expectation and variance of a random variable Let X be a random variable with the following probability distribution. Value x of X P(X-x) 0.05 5 0.25 6 0.15 7 0.55 Complete the following. (If necessary, consult a list of formulas.) (a) Find the expectation E (X ) of X. E(X) = 1 X 5 (b) Find the variance Var(X ) of X. Var (X) = Explanation CheckTheorem 1.4.11. If X is a discrete random variable with finite expectation and a is a constant, then ax has finite expectation and Efax] = aE[X]. If n 2 2 and each of the jointly distributed discrete random variables X1, . .., Xn has finite expectation, then so does X1+ ... + Xn and EXit . . . + X ] = E[Xi]+ . . . + E[X ]. (1.110)Theorem 1.4.17. If X is a discrete random variable with finite variance and a is a constant, then aX has finite variance and Var(aX ) = a2Var(X). If X1, X2, . .., Xn are jointly distributed discrete random variables, each with finite variance, then X1+ ... + Xn has finite variance and Var Exi = _Var(Xi) +2 > >Cov(Xi, X;). (1.127) i=1 1