AP Statistics Curriculum 2007 Gamma

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 \(\lambda\) 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 \(\theta=1/\lambda\), 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
• \(\textstyle\gamma(k,x/\theta)=\int_0^{x/\theta}t^{k-1}e^{-t}dt\)

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 \(\theta=1 / 0.5=2\) 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

Normal Approximation to Gamma distribution

Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent Exponential(b) 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 \(\sigma^2 =kb^2\). That is, the distribution of the variable \(Z_k={{Y_k-kb}\over{\sqrt{k}b}}\) tends to the standard normal distribution as \(k\longrightarrow \infty\).

For the example above, \(\Gamma(k=4, \theta=2)\), the SOCR Normal Distribution Calculator can be used to obtain an estimate of the area of interest as shown on the image below.

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The probabilities of the real Gamma and approximate Normal distributions (on the range [2:4]) are not identical but are sufficiently close.

Probability	\(\Gamma(k=4, \theta=2)\)	\(Normal(\mu=8, \sigma^2=4)\)
Mean	8.000000	8.0
Median	7.32	8.0
Variance	16.0	16.0
Standard Deviation	4.0	4.0
Max Density	0.112021	0.099736
<2	0.018988	0.066807
[2:4]	0.123888	0.091848
>4	0.857123	0.841345 \} SOCR Home page: http://www.socr.ucla.edu Translate this page: (default) Deutsch Español Français Italiano Português 日本語 България الامارات العربية المتحدة Suomi इस भाषा में Norge 한국어 中文 繁体中文 Русский Nederlands Ελληνικά Hrvatska Česká republika Danmark Polska România Sverige