# 12.3 - Poisson Properties

12.3 - Poisson Properties## Theorem

The probability mass function:

\(f(x)=\dfrac{e^{-\lambda} \lambda^x}{x!}\)

for a Poisson random variable \(X\) is a valid p.m.f.

### Proof

## Theorem

The moment generating function of a Poisson random variable \(X\) is:

\(M(t)=e^{\lambda(e^t-1)}\text{ for }-\infty<t<\infty\)

### Proof

## Theorem

The mean of a Poisson random variable \(X\) is \(\lambda\).

### Proof

## Theorem

The variance of a Poisson random variable \(X\) is \(\lambda\).