Mathematics for Computer Science

18.8. Really Great Expectations 659

(b)Conclude that PrŒonly finitely manyAn’s occurçD 0.

Hint:LetCkbe the event that noAnwithnkoccurs. So the event that only
finitely manyAn’s occur is
CWWD

[

k 2 N

Ck:

Apply part (a) toCk.

Problems for Section 18.8

Practice Problems

Problem 18.20.
LetRbe a positive integer valued random variable such that

PDFR.n/D

1

cn^3

;

where

cWWD

X^1

nD 1

1

n^3

:

(a)Prove that ExŒRçis finite.

(b)Prove that VarŒRçis infinite.

Problem 18.21.
LetTbe a positive integer valued random variable such that

PDFT.n/D

1

an^2

;

where

aWWD

X

n 2 ZC

1

n^2

:

(a)Prove that ExŒTçis infinite.

(b)Prove that ExŒ

p

Tçis finite.