AP Statistics Curriculum 2007 Gamma

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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(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.

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

Probability \(\Gamma(k=4, \theta=2)\) \(Normal(\mu=8, \sigma^2=4)\)
Mean8.0000008.0
Median7.328.0
Variance16.016.0
Standard Deviation4.04.0
Max Density 0.1120210.099736
<2 0.018988 0.066807
[2:4] 0.1238880.091848
>4 0.8571230.841345



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