AP Statistics Curriculum 2007 Gamma

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The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR distributions]
The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR distributions]
<center>[[Image:Gamma.jpg|600px]]</center>
<center>[[Image:Gamma.jpg|600px]]</center>
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===Normal Approximation to Gamma distribution===
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Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent [[AP_Statistics_Curriculum_2007_Exponential|Exponential random variables]] then \(Y_k = \sum_{i=1}^k{X_i} \) is a [http://www.math.uah.edu/stat/special/Gamma.html random variable with gamma distribution with some shape parameter], k (positive integer) and scale parameter b. By the [[AP_Statistics_Curriculum_2007_Limits_CLT|central limit theorem]], if k is large, then gamma distribution can be approximated by the normal distribution with mean \(\mu=kb\) and variance \(\sigma =kb^2\). That is, the distribution of the variable <math>Z_k=\fract{Y_k-kb}{\sqrt{k}b}</math> tends to the standard normal distribution as <math>k\longrightarrow \infty</math>.
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Revision as of 16:55, 23 June 2012

Contents

General Advance-Placement (AP) Statistics Curriculum - Gamma Distribution

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\sim \operatorname{Gamma}(k,\theta)\!, 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
  • \textstyle\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


Normal Approximation to Gamma distribution

Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent Exponential random variables then \(Y_k = \sum_{i=1}^k{X_i} \) is a random variable with gamma distribution with some shape parameter, k (positive integer) and scale parameter b. By the central limit theorem, if k is large, then gamma distribution can be approximated by the normal distribution with mean \(\mu=kb\) and variance \(\sigma =kb^2\). That is, the distribution of the variable Failed to parse (unknown function\fract): Z_k=\fract{Y_k-kb}{\sqrt{k}b}

tends to the standard normal distribution as k\longrightarrow \infty.



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