Question

In each game played one is equally likely to either win or lose 1. Let X be your cumulative winnings if you use the strategy that quits playing if you win the first game, and plays two more games and then quits if you lose the first game.

(a) Use Wald’s equation to determine E[X].

(b) Compute the probability mass function of X and use it to find E[X].

Answer 1

a) Wald's equation is given by:

, here while , thus

E[X] -= 2*0.5 =1

b) The pmf of X is:

case 1: wins first game with probability 0.5. Thus X=1 w.p. 0.5

case 2: loses the first game with probability. Thus X can be 0,1,2 and X follows a binomial(2,0.5).

Thus pmf of X adding the two cases is:

X=x	0	1	2
P(X=x)	0.125	0.75	0.125

Here P(X=0) = P(X=0| loses first game)*P(loses first game) = 0.25*0.5=0.125

P(X=1) = P(X=1| loses first game)*P(loses first game) + P(X=1|wins first game)*P(wins first game)= 0.5*0.5+ 1*0.5 = 0.75

P(X=2) = P(X=2| loses first game)*P(loses first game) = 0.25*0.5=0.125

Hence E[X] = 0*0.125 + 1*0.75 + 2*0.125 = 1