Poisson distribution

If $x\in \{0,1,2,...\}$ has Poisson distribution with parameter $\lambda >0$ then the probability mass function is

$$\text{Poisson}(x;\lambda )=\mathrm{Pr}(X=x)=\frac{{\lambda }^x{e}^{-\lambda }}{x!}$$

We write $x|\lambda \sim \text{Poisson}(\lambda )$ and $\pi (x|\lambda )=\text{Poisson}(x;\lambda )$.