13. Let X be a discrete random variable. (a) Prove that if mjnjaC for all n, then PrjX mjbt 0 for any t

13. Let X be a discrete random variable.

(a) Prove that if mjnjaC for all n, then Pr½jX  mjbt ¼ 0 for any t > 1. In other words, it must be the case that Pr½jX  mja1 ¼ 1: (Hint: Chebyshev.)

(b) Generalize part (a). Prove that if mjnjaCn for all n, then Pr½jX  mjbt ¼ 0 for any t > C.

(c) Conclude that if X has unbounded range, then it cannot be the case that mjnjaCn for any C.