AP Statistics Curriculum 2007 Gamma
From Socr
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 λ is
![P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}](/distributome/uploads/math/9/e/e/9eeca539727423c1e3ab7dcdd177552e.png)
For X~Gamma(k,θ), where k = h and θ = 1 / λ, the gamma probability density function is given by
![\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}](/distributome/uploads/math/4/b/3/4b3ccb9a94ff3c8793161d9933716458.png)
where
- e is the natural number (e = 2.71828…)
- k is the number of occurrences of an event
- if k is a positive integer, then Γ(k) = (k − 1)! is the gamma function
- θ = 1 / λ is the mean number of events per time unit, where λ is the mean time between events. For example, if the mean time between phone calls is 2 hours, then you would use a gamma distribution with θ=1/2=0.5. If we want to find the mean number of calls in 5 hours, it would be 5
1/2=2.5.
- x is a random variable
Cumulative density function: The gamma cumulative distribution function is given by
![\frac{\gamma(k,x/\theta)}{\Gamma(k)}](/distributome/uploads/math/7/8/3/7836002a6f62dd7fba39da999524d5a2.png)
where
- if k is a positive integer, then Γ(k) = (k − 1)! is the gamma function
Moment generating function: The gamma moment-generating function is
![M(t)=(1-\theta t)^{-k}\!](/distributome/uploads/math/3/e/5/3e53f29c69958a696306f5a4c2ed25cd.png)
Expectation: The expected value of a gamma distributed random variable x is
![E(X)=k\theta\!](/distributome/uploads/math/9/6/2/962abe26e128cf2c5c13c9bf3e9d1646.png)
Variance: The gamma variance is
![Var(X)=k\theta^2\!](/distributome/uploads/math/9/d/a/9da98faf651bc9d799e3ba1f48e7127b.png)
Applications
The gamma distribution can be used a range of disciplines including queuing models, climatology, and financial services. Examples of events that may be modeled by gamma distribution include:
- The amount of rainfall accumulated in a reservoir
- The size of loan defaults or aggregate insurance claims
- The flow of items through manufacturing and distribution processes
- The load on web servers
- The many and varied forms of telecom exchange
The gamma distribution is also used to model errors in a multi-level Poisson regression model because the combination of a Poisson distribution and a gamma distribution is a negative binomial distribution.
Example
Suppose you are fishing and you expect to get a fish once every 1/2 hour. Compute the probability that you will have to wait between 2 to 4 hours before you catch 4 fish.
One fish every 1/2 hour means we would expect to get θ = 1 / 0.5 = 2 fish every hour on average. Using θ = 2 and k = 4, we can compute this as follows:
The figure below shows this result using SOCR distributions
![](/distributome/uploads/thumb/1/1b/Gamma.jpg/600px-Gamma.jpg)