Answer Happy

Suppose that {Xn} is a sequence of random variables such that Xn +d Xwhere E(X) is finite. We would like to investigate sufficient conditions under which E(Xn) → E(X) (assuming that E(X) is well-defined). this convergence holds if the Xn's are uniformly bounded. Hint: Use the fact that M .” | P(X.) > 2) – P(|X|> 2) dx + 0 as n – for each finiteM. 0

(b) Show that for any M > 0 and 8 > 0 M P(|Xn] > x)dx = E(|Xn]) 6. "/ )E S“ P(X.) > a)dz + S.* 2*P(X./ > )dr M 00 < M8 M Again let 8 > 0 and suppose that E(|Xn|1+0) SK < @ for all n. Assuming that Xn +d X, use the results of parts (a) and (b) to show that E(|Xn]) + E(X) and E(Xn) → E(X) as n +0.