AP Statistics Curriculum 2007 Gamma

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(Gamma Distribution)
(Gamma Distribution)
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*k is the number of occurrences of an event
*k is the number of occurrences of an event
*if k is a positive integer, then <math>\Gamma(k)=(k-1)!</math> is the gamma function  
*if k is a positive integer, then <math>\Gamma(k)=(k-1)!</math> is the gamma function  
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*<math>\theta=1/\lambda</math> is the mean number of events per time unit, where <math>\lambda</math> 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 <math>\theta</math>=1/2=0.5. If we want to find the mean number of calls in 5 hours, it would be 5 <math>\times</math> 1/2=2.5.
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*<math>\theta=1/\lambda</math> is the mean number of events per time unit, where <math>\lambda</math> 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 <math>\theta</math>=1/2=0.5. If we want to find the mean number of calls in 5 hours, it would be <math>5\times 1/2=2.5</math>.
*x is a random variable
*x is a random variable

Revision as of 20:25, 11 July 2011

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}}


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}

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\times 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)}

where

  • if k is a positive integer, then Γ(k) = (k − 1)! is the gamma function
  • \gamma(k,x/\theta)=\int_0^{x/\theta}t^{k-1}e^{-t}dt


Moment generating function: The gamma moment-generating function is

M(t)=(1-\theta t)^{-k}\!


Expectation: The expected value of a gamma distributed random variable x is

E(X)=k\theta\!


Variance: The gamma variance is

Var(X)=k\theta^2\!

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:

P(2\le X\le 4)=\sum_{x=2}^4\frac{x^{4-1}e^{-x/2}}{\Gamma(4)2^4}=0.12388

The figure below shows this result using SOCR distributions

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