# AP Statistics Curriculum 2007 Gamma

### Gamma Distribution

Definition: **Gamma distribution** is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events.

Probability density function: The waiting time until the hth Poisson event with a rate of change \(\lambda\) is

\begin{center} \(P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}\) end{center}

For **X~Gamma(k,\(\theta\))**, where \(k=h\) and \(\theta=1/\lambda\), the gamma probability density function is given by

\(\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}\)

For **X~Poisson(λ)**, the Poisson mass function is given by \(P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!\) where

*e*is the natural number (*e*= 2.71828...)*k*is the number of occurrences of an event - the probability of which is given by the mass function- \(k! = 1\times 2\times 3\times \cdots \times k\)
- λ is a positive real number, equal to the
*expected number of occurrences*that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ=10/4=2.5.