Question

Exercise 3.52. The idea in Exercise 3.51 generalizes to give a new formula for the expected value of any nonnegative integer-valued random variable. Show that if the random variable X takes only nonnegative integers as its values then に! 00, in which case the sum on the right-hand side This holds even when E(X) is infinite. Hint. Write P(X > k) as Σ_kPX = ) in the sum, and then switch the order of the two summations.

Answer 1

E(x) = sigma_x = 0^infinity x p(x) = op(x = 0) + lp(x = 1) + 2p(x = 2) + 3p(x = 3) + ---- = p(x = 1) + p(x = 2) + p(x = 3) + p(x = 4) + ---- = p(x greaterthanequalto 1) + p(x greaterthanequalto 2), + p(x greaterthanequalto 3) + ---- E(x) = sigma_x = 1^infinity p(x greaterthanequalto u) let us assume that E(x) = infinity in this case we can say that p(x = i) = ti forall it Z^+ V{0} and Ei lessthanorequalto 0 also using the addition prop of prob we have sigma_i = 0^infinity p(x = i) = sigma_i = 1^infinity ti = j but here E(x) = sigma_i = 1^infinity p(x>i) = sigma_j = 0^infinity sigma_i = 0^infinity tij = infinity