# Difference between revisions of "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. 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