(Refer to exercise 3.2 for this question, do not answer 3.2) Calculate the mean

Question

(Refer to exercise 3.2 for this question, do not answer 3.2) Calculate the mean and variance of Y, your winnings on a single play of the game. Note that E(Y) greater than 0. How much should you pay to play this game if your net winnings, the difference between the payoff and cost of playing, are to have mean 0?

(exercise 3.2)

You and a friend play a game where you each toss a balanced coin. If the upper fces on the coins are both tails, you win $1: if the faces are both heads, you win $2; if the coins do not match (one shows a head, the other a tail), you lose $1( win (-$1)). Give the probability distribution for you winnings, Y, on a single play of this game.

Explanation / Answer

This is very simple. You only have 4 possible outcomes, each outcome has probability 1/4:

tails-tails => +$1
heads-heads => +$2
tails-heads => -$1
heads-tails => -$1

So, your winnings (Y) can have 3 values: +1, +2 and -1. Each of the first two has probability 1/4. And the probability that Y=-1 is 2/4=1/2 (because this happens in two outcomes).

So, the answer:
y: -1 +1 +2
P(Y=y):1/2 1/4 1/4

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