Question

The following gambling game has been proposed, which a player must pay to play. First, a value U is chosen uniformly from the set [0, 10]. Next, a number is chosen according to a Poisson random variable with a parameter U. Letting X be the number chosen, the player receives $X. Find E[X], which is the amount a player should pay to make this a fair game HINT: Use the Law of Total Probability for Expectations, E[X]

Answer 1

The player should be required to pay $5 to make this a fair game.

Explanation:

U ~ Uniform(0, 10)

E[U] = (0 + 10)/2

= 5

X | U ~ Poisson(U)

E[X | U] = U

By law of total probability for expectations,

E[X] = E[E[X|U]] = E[U] = $5

Therefore the player should be required to pay $5 to make this a fair game.