# Poisson Distribution and slot machine

## Homework Statement

A casino slot machine costs C dollars per play. On each play, it generates random variable X ~ Poisson with parameter λ < 1, and pays the player X! (X factorial) dollars. As a function of the fixed parameters λ and C, how much money would you expect to win (or lose) per turn if you play? How much should the casino operators charge so that they don't lose money, i.e., what value of C should they use for a fixed λ? What if λ=1? Hint: E[X!] is not equal to (E[X])! in general.

## The Attempt at a Solution

I am at a complete loss of how to solve this problem. The only think I can think of is that

a) How much money would you expect to win or lose per turn
(e^-λ)! - C

b) So the casino players should charge C = (e^-λ)!

I am really very confused with this question. Any help to point me to the correct direction of thinking would be appreciated. Thank you.

I don't have an answer, but here is an observation: the expectation of X! is
$$\mathbb{E}(X!)=\sum_{k=0}^{\infty}k!\cdot Pr(k\;|\;\lambda)=$$
$$= \sum_{k=0}^{\infty}k!\cdot\dfrac{\lambda^k e^{-\lambda}}{k!}=e^{-\lambda}\sum_{k=0}^{\infty}\lambda^k$$

If you know how this sum can be expressed, you got the expected winning sum.

Regards,
Joseph.