# AP Statistics Curriculum 2007 Gamma

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### 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.