Let $X \in \text{Poisson}(\theta)$ be a D2854: Poisson random natural number.

By definition, $e^x : = \sum_{n = 0}^{\infty} x^n / n !$ for all real numbers $x \in \mathbb{R}$. Thus, applying R1814: Expectation of discrete random euclidean real number, we have \begin{equation} \begin{split} \mathbb{E}(X) & = \sum_{n = 0}^{\infty} n \mathbb{P}(X = n) \\ & = \sum_{n = 1}^{\infty} n \mathbb{P}(X = n) \\ & = e^{- \theta} \sum_{n = 1}^{\infty} n \frac{\theta^n}{n!} \\ & = \theta e^{- \theta} \sum_{n = 1}^{\infty} \frac{\theta^{n - 1}}{(n - 1)!} \\ & = \theta e^{- \theta} \sum_{n = 0}^{\infty} \frac{\theta^n}{n!} \\ & = \theta e^{- \theta} e^{\theta} \\ & = \theta \end{split} \end{equation} $\square$