Question

Problem 3.4 (10 points) Consider this game of chance with a monetary payoff. First, a real number is chosen uniformly at random from the interval [0,10]. Next, an integer X is chosen according to the Poisson distribution with parameter U. The player receives a reward of SX What would be the fair price charged for playing this game? That is, how much should it cost to play so that expected net gain is zero?

Answer 1

U has a uniform distribution in the interval [0,10]

X has a Poisson distribution with parameter U. We have the following hierarchy

$\begin{align*} &X\mid U \sim \text{Poisson}(U)\\ &U\sim \text{Uniform}(0,10) \end{align*}$

The expectation of U is

$\begin{align*} E(U)=\frac{a+b}{2}=\frac{0+10}{2}=5 \end{align*}$

The conditional expectation of X given U is

$\begin{align*} E(X\mid U=u)=u \end{align*}$

The expected value of X is

$\begin{align*} E(X)&=E[E(X\mid U)]\quad\text{using the formula for iterated expectations}\\ &=E(U)\\ &=5 \end{align*}$

The expected value of reward received by the player is $5

Let c be the price charged for playing the game.

If Y is the net gain, the net gain is given by

Y=X-c

The expected net gain is

$\begin{align*} E(Y)&=E(X-c)\\ &=E(X)-c\quad\text{using the result for any constant a,b }E(aX+b)=aE(X)+b\\ &=5-c \end{align*}$

For the expected net gain to be 0, the value of c=5.

Ans: The fair price charged for playing this game is $5.