Discrete Probability Distribution

Poisson Distribution

We use the Poisson Distribution to answer the following question:

• Given a rate $\lambda$, what is the probability of $x$ successes, i.e. $P(X=x)$?

The Poisson Distribution was invented to predict the probability of a given number of events in a fixed interval of time.

Understand the Poisson Process.

title: Poisson Distribution (Definition) 
We say $X ∼ Poi(λ)$, where
$$P(X = x) = f(x) = \frac{e^{-\lambda}\lambda^x}{x!} , \set{x = 0, 1, 2, . . .,} \text{ and } \lambda > 0$$
 
where
- $\lambda$ is Rate/mean, or average number of successes/failures per unit

The support of $X$ is $\{0,1,2,...\}$.

• $E(X)=\lambda$
• $Var(X)=\lambda$

title:Modelling
Poisson models are useful in modelling events where N is large, and p is small, and we are interested in the number of successes (or arrivals).