Mean Poisson distribution = $\lambda$

Poisson Distribution
\begin{eqnarray*} P(X=x) = \frac{e^{-\lambda} \cdot \lambda^x}{x!} \\ \end{eqnarray*}
혹은
\begin{eqnarray*} P(x) = \frac{e^{-\lambda} \cdot \lambda^x}{x!} \\ \end{eqnarray*}

우선 Taylor series을 이용하면
\begin{eqnarray*} e^{a} = \sum_{y=0}^{\infty} \frac{a^y}{y!} \\ \end{eqnarray*}
임을 알고 있다.

\begin{eqnarray*} E(X) & = & \sum_{x} xp(X=x) \\ \text{or } \\ E(X) & = & \sum_{x} xp(x) \\ \end{eqnarray*}
Poisson distribution 을 다루고 있으므로
\begin{eqnarray*} E(X) & = & \sum_{x=0}^{\infty} x \cdot \frac{e^{-\lambda} \cdot \lambda^x}{x!} \\ & = & e^{-\lambda} \cdot \sum_{x=0}^{\infty} x \cdot \frac{\lambda^x}{x!} \\ & = & e^{-\lambda} \cdot \sum_{x=0}^{\infty} x \cdot \frac{\lambda^x}{x(x-1)!} \\ & = & e^{-\lambda} \cdot \sum_{x=1}^{\infty} x \cdot \frac{\lambda \cdot \lambda^{x-1}}{x(x-1)!} \\ & = & \lambda \cdot e^{-\lambda} \cdot \sum_{x=1}^{\infty} \frac{\lambda^{x-1}}{(x-1)!} \\ \text{let y = x-1} & \\ & = & \lambda \cdot e^{-\lambda} \cdot \sum_{y=0}^{\infty} \frac{\lambda^{y}}{(y)!} \\ & = & \lambda \cdot e^{-\lambda} \cdot \underline{\sum_{y=0}^{\infty} \frac{\lambda^{y}}{(y)!}} \\ \text{recall: } \; \\ e^{a} = \sum_{y=0}^{\infty} \frac{a^y}{y!} \\ & = & \lambda \cdot e^{-\lambda} \cdot e^{\lambda} \\ & = & \lambda \\ \end{eqnarray*}

Variance는 위의 binomial 케이스처럼 좀 복잡하다.
Variance of Poisson distribution = $\lambda$