12.3 - Poisson Properties
12.3 - Poisson PropertiesJust as we did for the other named discrete random variables we've studied, on this page, we present and verify four properties of a Poisson random variable.
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 \(\lambda\).