Poisson Likelihood

$$X\sim \text{Poisson}(\lambda ),\lambda >0$$

PMF

$$P(X=x)=\frac{e^{-\lambda }{\lambda }^x}{x!},x=0,1,2,\dots$$

Likelihood (single observation)

$$L(\lambda \mid x)=\frac{e^{-\lambda }{\lambda }^x}{x!}$$

Likelihood (i.i.d. sample)
Let $X_1,\dots ,X_n\sim \text{Poisson}(\lambda )$ and $S={\sum }_{i=1}^n{X}_i$.

$$L(\lambda \mid x_1,\dots ,x_n)=e^{-n\lambda }{\lambda }^S\prod _{i=1}^n\frac{1}{x_i!}$$

Ignoring constants:

$$L(\lambda )\propto e^{-n\lambda }{\lambda }^S$$

Log-Likelihood

$$\ell (\lambda )=S\log \lambda -n\lambda +C$$