Expectation of Poisson Distribution/Proof 1

Proof
From the definition of expectation:


 * $\ds \expect X = \sum_{x \mathop \in \Img X} x \map \Pr {X = x}$

By definition of Poisson distribution:


 * $\ds \expect X = \sum_{k \mathop \ge 0} k \frac 1 {k!} \lambda^k e^{-\lambda}$

Then: