What is Poisson Distribution?

What is Poisson Distribution?

Poisson random variables are used to measure counts of events. Poisson random variables are always non-negative and integer-valued. The Poisson Distribution has a single parameter called the hazard rate and is expressed as λ, which signifies the average number of events per interval. Therefore, the mean and variance of Y ~ Poisson(λ) are:

E[Y] = V[Y] = λ

PMF of Poisson:

$ f(Y)y= \frac{\lambda ^{y}exp(-\lambda)}{y!} $

CDF of Poisson:

Meanwhile, the CDF of a Poisson is defined as the sum of the PMF for values less than the input.

$ F(Y)_{y}= exp(-\lambda )\sum_{i=0}^{y}\frac{\lambda ^{i}}{i!} $

Why is Poisson Distribution important?

A Poisson distribution is a tool that helps to predict the probability of certain events happening when you know how often the event has occurred. It gives us the likelihood of a given number of events happening in a fixed time interval. One application of a Poisson is to model the number of loan defaults that occur each month.