Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 846 — #854
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Chapter 19
Random Variables
Problem 19.26.
Justify each line of the following proof that if R 1 and R 2 are independent, then
Proof.
ExŒR 1 R 2
X
D
D
D
D
D
D
r2range.R 1 R 2 /
X
r i 2range.R i /
X
ExŒR 1 R 2 D ExŒR 1 ExŒR 2 :
r PrŒR 1 R 2 D r
r 1 r 2 PrŒR 1 D r 1 and R 2 D r 2
X
r 1 2range.R 1 / r 2 2range.R 2 /
X
X
r 1 2range.R 1 / r 2 2range.R 2 /
X
r 1 2range.R 1 /
X
r 1 2range.R 1 /
D ExŒR 2
0
@r 1 PrŒR 1 D r 1
r 1 r 2 PrŒR 1 D r 1 and R 2 D r 2
r 1 r 2 PrŒR 1 D r 1 PrŒR 2 D r 2
r 1 PrŒR 1 D r 1 ExŒR 2
X
r 1 2range.R 1 /
D ExŒR 2 ExŒR 1 :
r 1 PrŒR 1 D r 1
X
r 2 2range.R 2 /
1
r 2 PrŒR 2 D r 2 A
Problem 19.27.
A gambler bets on the toss of a fair coin: if the toss is Heads, the gambler gets back
the amount he bet along with an additional the amount equal to his bet. Otherwise
he loses the amount bet. For example, the gambler bets $10 and wins, he gets back
$20 for a net profit of $10. If he loses, he gets back nothing for a net profit of
$10—that is, a net loss of $10.
Gamblers often try to develop betting strategies to beat the odds is such a game.
A well known strategy of this kind is bet doubling, namely, bet $10 on red, and
keep doubling the bet until a red comes up. So if the gambler wins his first $10
bet, he stops playing and leaves with his $10 profit. If he loses the first bet, he bets