You toss independently a fair coin and you count the number of tosses until the first tail appears. If this number is n, you receive 2n dollars. What is the expected amount that you will receive? How much would you be willing to pay to play this game?

Step-by-step explanation:

a)

If you toss independently a fair coin and you count the number of tosses until the first tail appears.

This is a geometric distribution with (win or lose)

The probability mass function is,

The payout is and this payout has the same PMF as X

The expected value of the Y

b)

If you are faced with the choice of playing for give fee or not playing at all and your objective is to make the choice that maximize your expected net gain. You would be willing to pay any value of fee. However, this is in strong disagreement with the behaviour of individuals. The discrepancy is due to a presumption that the amount one is willing to pay is determined by the expected gain.

the image of (6, 9) under a dilation is (4, 6). the scale factor is

the answer is 99.