Difference between revisions of "AP Statistics Curriculum 2007 Gamma"

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=== Gamma Distribution ===
 
=== Gamma Distribution ===
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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.
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'''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.
  
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Probability density function: The waiting time until the hth Poisson event with a rate of change <math>\lambda</math> is
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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
  
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\begin{center} <math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math> end{center}
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:<math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math>
  
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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
 
  
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<math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math>
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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>
  
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where
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*e is the natural number (e = 2.71828…)
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*k is the number of occurrences of an event
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*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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*x is a random variable
  
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<br />'''Cumulative density function''': The gamma cumulative distribution function is given by
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:<math>\frac{\gamma(k,x/\theta)}{\Gamma(k)}</math>
  
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For '''X~Poisson(λ)''', the Poisson mass function is given by <math>P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!</math> where
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where
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** ''e'' is the [http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 natural number] (''e'' = 2.71828...)
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*if k is a positive integer, then <math>\Gamma(k)=(k-1)!</math> is the gamma function  
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** ''k'' is the number of occurrences of an event - the probability of which is given by the mass function
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*<math>\gamma(k,x/\theta)=\int_0^{x/\theta}t^{k-1}e^{-t}dt</math>
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** <math>k! = 1\times 2\times 3\times \cdots \times k</math>
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** 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.
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<br />'''Moment generating function''': The gamma moment-generating function is
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:<math>M(t)=(1-\theta t)^{-k}\!</math>
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<br />'''Expectation''': The expected value of a gamma distributed random variable x is  
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:<math>E(X)=k\theta\!</math>
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<br />'''Variance''': The gamma variance is  
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:<math>Var(X)=k\theta^2\!</math>
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 +
 
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===Applications===
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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:
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*The amount of rainfall accumulated in a reservoir
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*The size of loan defaults or aggregate insurance claims
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*The flow of items through manufacturing and distribution processes
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*The load on web servers
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*The many and varied forms of telecom exchange
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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.
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 +
 
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===Example===
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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.
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 +
One fish every 1/2 hour means we would expect to get <math>\theta=1/0.5=2</math> fish every hour on average. Using <math>\theta=2</math> and <math>k=4</math>, we can compute this as follows:
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:<math>P(2\le X\le 4)=\sum_{x=2}^4\frac{x^{4-1}e^{-x/2}}{\Gamma(4)2^4}=0.12388</math>
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 +
The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR distributions]

Revision as of 13:31, 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

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