Poisson Distribution

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

This holds according to Taylor series.

$$E(X)=\text{Var}(X)=\lambda$$

The Poisson distribution is often used in situations where we are counting the number of successes in a particular region or interval of time, and there are a large number of trials, each with a small probability of success.

$\lambda$ is the rate of the rare events.

The sum of independent Poissons is Poisson — proved by LOTP.

Poisson Approximation

Poisson approximation for small $p_j$s, aka the law of rare events.

$$X=\sum _{j=1}^{n}I(A_j)$$

Poisson approximation for Birthday Problem.

$$\lambda =\left(\genfrac{}{}{0ex}{}{m}{2}\right)\frac{1}{365}$$