Week 6

Definition of Expectation.

The expectation definition for the Poisson PMF follows.
\[E(X)=\sum_0^x x\frac {e^{-\lambda} \lambda^x}{x!}, x \in X \ge 0\]

Reduce

\[E(X)=\sum_0^{x-1} \frac {e^{-\lambda} \lambda^x}{(x-1)!}\]

Extract

\[E(X)=\lambda \times \sum_0^{x-1} \frac {e^{-\lambda} \lambda^{x-1}}{(x-1)!}\]

Definition of Poisson PMF and Properties of PMF

The part to the right of \(\lambda\) is the PMF of another Poisson. We know that this PMF sums to 1 by definition of the PMF.

\[E(X)=\lambda \times 1=\lambda\]