AP Statistics Curriculum 2007 Gamma
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General AdvancePlacement (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
For , where k = h and θ = 1 / λ, the gamma probability density function is given by
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 1/2=2.5.
 x is a random variable
Cumulative density function: The gamma cumulative distribution function is given by
where
 if k is a positive integer, then Γ(k) = (k − 1)! is the gamma function
Moment generating function: The gamma momentgenerating function is
Expectation: The expected value of a gamma distributed random variable x is
Variance: The gamma variance is
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 multilevel 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:
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 \(Z_k={{Y_kkb}\over{\sqrt{k}b}}\) tends to the standard normal distribution as .
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.
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
\}
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