. (Pitman 3.4.9) Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n 2 if heads comes up first on the nth toss. If we play this game repeatedly, how much money do you expect to win or lose per game over the long run

Question

Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n² if heads comes up first on the nth toss. If we play this game repeatedly, how much money do you expect to win or lose per game over the long run

Lose of $4 per game

Explanation:

Let X represents the toss on which the first head appears.

Then X has a geometric distribution with parameter

p = ½

q = 1 - p

q = 1 - ½

q = ½

and you expect to be paid

A =∞ to n=1 for n² * P(X=n) =E(X²)

Where E(X²) = (1 + q)/p²

E(X²) = (1 + ½)/½²

= 3/2 ÷ ¼

= 3/2 * 4/1

= 12/2

= 6

Since you pay $10 to play the game,

you expect to lose $10 - $6 per game.

= $4