Another important distribution with discrete stochastic variables $x$ is the Poisson model, which resembles the exponential distribution and reads $$\begin{equation*} p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. \end{equation*}$$ In this case both the mean value and the variance are easier to calculate, $$\begin{equation*} \mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} \frac{\lambda^{x-1}}{(x-1)!}=\lambda, \end{equation*}$$ and the variance is $\sigma^2=\lambda$.