5.47 (i) Show that for any random variable X andp > 0, we have E{Xp1(X≥0)} =

 ∞

0 pxp−1P(X ≥ x) dx.

[Hint: Note that Xp1(X≥0) =



X 0 pxp−11(X≥0) dx =

∞

0 pxp−11(X≥x) dx.

Use the result in Appendix A.2 to justify the exchange of order of expectation and integration.]

(ii) Show that if X1, . . . , Xn are independent and symmetrically distributed about zero, then for any p > 0, E



max 1≤m≤n Sm



p 1(max1≤m≤n Sm≥0)

≤ 2E{S p

n 1(Sn≥0)}, where Sm =



n i=1 Xi .