Question

What is the difference between Poisson Distribution and Exponential Distribution?

Answer 1

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(y)=(y^(alpha-1)e^(-y)/(beta))/(beta^(alpha)Gamma(alpha))`

and zero elsewhere, where `Gamma(alpha)` is the gamma function with argument `alpha`.

`Gamma(alpha)=int_0^(oo)y^(alpha-1)e^(-y)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(y)=1/(beta)e^(-y/beta)," "0 <= y < oo`

and zero elsewhere.

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