Poisson RV

A discrete RV $X$ with range $\mathscr{X}=\{1,\cdots\}$ is called a Poisson random variable with parameter $\lambda>0$ if its pmf is given by $$p(i)=\frac{e^{-\lambda}\lambda^i}{i!}$$ For RV $X\sim \mbox{Poisson}(\lambda)$$$\begin{align*} E[X]&=\lambda\\ Var(X)&=\lambda \end{align*}$$