# What is the difference between Poisson Distribution and Exponential Distribution?

Dec 16, 2017

The Poisson distribution models "rare" events; the exponential distribution models distributions of data that are skewed to the right.

#### Explanation:

The Poisson probability distribution often provides a good model for the probability distribution of the number of $Y$ "rare" events that occur in space, time, volume, or any other dimension.

Examples include car/industrial accidents, telephone calls handled by a switchboard in a time interval, number of radioactive particles that decay in a particular time period, etc.

A random variable $Y$ is said to have a Poisson probability distribution if and only if

p(y)=(lambda^y)/(y!)e^(-lambda)" "y=0,1,2,...,lambda>0

Where $\lambda$ is the average value of $Y$.

The exponential probability distribution is actually a specific case of the gamma probability distribution.

The gamma density function does a sufficient job of modeling the populations associated with random variables that are always nonnegative and yield distributions of data that are skewed (non symmetric) to the right.

A random variable $Y$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if and only if the density function of $Y$ is

$f \left(y\right) = \frac{{y}^{\alpha - 1} {e}^{- y} / \left(\beta\right)}{{\beta}^{\alpha} \Gamma \left(\alpha\right)}$

and zero elsewhere, where $\Gamma \left(\alpha\right)$ is the gamma function with argument $\alpha$.

$\Gamma \left(\alpha\right) = {\int}_{0}^{\infty} {y}^{\alpha - 1} {e}^{- y} \mathrm{dy}$

A random variable $Y$ is said to have an exponential distribution with parameter $\beta > 0$ if and only if the density function of $Y$ is

$f \left(y\right) = \frac{1}{\beta} {e}^{- \frac{y}{\beta}} , \text{ } 0 \le y < \infty$

and zero elsewhere.

Essentially, the exponential distribution is the gamma distribution, just with $\alpha = 1$ and $\beta = \beta$. Note that Gamma(1)=0! =1.