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.

Poisson Distribution (Definition)

We say $X\sim Poi(\lambda )$, where $P(X=x)=f(x)=\frac{e^{-\lambda }{\lambda }^x}{x!},\{x=0,1,2,...,\}\text{ and }\lambda >0$

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).

Practice this:

• https://www.deep-ml.com/problems/81

A more intuitive rephrasing is $P(k,\lambda )=\frac{e^{-\lambda }{\lambda }^k}{k!}$ ​

• ( k ): Number of events (non-negative integer)
• ( \lambda ): The mean number of events in the given interval (rate parameter)
• ( e ): Euler’s number, approximately 2.718