Difference between revisions of "AP Statistics Curriculum 2007 Gamma"

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(Example)
(Gamma Distribution)
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<br />'''Probability density function''': The waiting time until the hth Poisson event with a rate of change <math>\lambda</math> is
 
<br />'''Probability density function''': The waiting time until the hth Poisson event with a rate of change <math>\lambda</math> is
  
:<math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math>
+
<center><math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math></center>
  
  
 
For X~Gamma(k,<math>\theta</math>), where <math>k=h</math> and <math>\theta=1/\lambda</math>, the gamma probability density function is given by
 
For X~Gamma(k,<math>\theta</math>), where <math>k=h</math> and <math>\theta=1/\lambda</math>, the gamma probability density function is given by
  
:<math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math>
+
<center><math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math></center>
  
 
where
 
where
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<br />'''Cumulative density function''': The gamma cumulative distribution function is given by  
 
<br />'''Cumulative density function''': The gamma cumulative distribution function is given by  
:<math>\frac{\gamma(k,x/\theta)}{\Gamma(k)}</math>
+
<center><math>\frac{\gamma(k,x/\theta)}{\Gamma(k)}</math></center>
  
 
where
 
where
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<br />'''Moment generating function''': The gamma moment-generating function is  
 
<br />'''Moment generating function''': The gamma moment-generating function is  
:<math>M(t)=(1-\theta t)^{-k}\!</math>
+
<center><math>M(t)=(1-\theta t)^{-k}\!</math></center>
  
 
<br />'''Expectation''': The expected value of a gamma distributed random variable x is  
 
<br />'''Expectation''': The expected value of a gamma distributed random variable x is  
:<math>E(X)=k\theta\!</math>
+
<center><math>E(X)=k\theta\!</math></center>
  
 
<br />'''Variance''': The gamma variance is  
 
<br />'''Variance''': The gamma variance is  
:<math>Var(X)=k\theta^2\!</math>
+
<center><math>Var(X)=k\theta^2\!</math></center>
 
 
  
 
===Applications===
 
===Applications===

Revision as of 14:37, 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 \(\lambda\) is

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


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

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

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 \(\Gamma(k)=(k-1)!\) is the gamma function
  • \(\theta=1/\lambda\) is the mean number of events per time unit, where \(\lambda\) 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 \(\theta\)=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 \(\Gamma(k)=(k-1)!\) 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 \(\theta=1/0.5=2\) fish every hour on average. Using \(\theta=2\) 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

Gamma.jpg