AP Statistics Curriculum 2007 Gamma
From Socr
(Created page with '=== Gamma Distribution === Definition: '''Gamma distribution''' is a distribution that arises naturally in processes for which the waiting times between events are relevant. It c…') |
(→Gamma Distribution) |
||
Line 1: | Line 1: | ||
=== 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 <math>\lambda</math> is | + | <br />'''Probability density function''': The waiting time until the hth Poisson event with a rate of change <math>\lambda</math> is |
- | + | :<math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math> | |
- | |||
- | <math>\ | + | For X~Gamma(k,<math>\theta</math>), where <math>k=h</math> and <math>\theta=1/\lambda</math>, the gamma probability density function is given by |
+ | :<math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math> | ||
+ | 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 <math>\Gamma(k)=(k-1)!</math> is the gamma function | ||
+ | *<math>\theta=1/\lambda</math> is the mean number of events per time unit, where <math>\lambda</math> 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 <math>\theta</math>=1/2=0.5. If we want to find the mean number of calls in 5 hours, it would be 5 <math>\times</math> 1/2=2.5. | ||
+ | *x is a random variable | ||
+ | <br />'''Cumulative density function''': The gamma cumulative distribution function is given by | ||
+ | :<math>\frac{\gamma(k,x/\theta)}{\Gamma(k)}</math> | ||
- | + | where | |
- | + | *if k is a positive integer, then <math>\Gamma(k)=(k-1)!</math> is the gamma function | |
- | + | *<math>\gamma(k,x/\theta)=\int_0^{x/\theta}t^{k-1}e^{-t}dt</math> | |
- | + | ||
- | + | <br />'''Moment generating function''': The gamma moment-generating function is | |
+ | :<math>M(t)=(1-\theta t)^{-k}\!</math> | ||
+ | |||
+ | <br />'''Expectation''': The expected value of a gamma distributed random variable x is | ||
+ | :<math>E(X)=k\theta\!</math> | ||
+ | |||
+ | <br />'''Variance''': The gamma variance is | ||
+ | :<math>Var(X)=k\theta^2\!</math> | ||
+ | |||
+ | |||
+ | ===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 <math>\theta=1/0.5=2</math> fish every hour on average. Using <math>\theta=2</math> and <math>k=4</math>, we can compute this as follows: | ||
+ | :<math>P(2\le X\le 4)=\sum_{x=2}^4\frac{x^{4-1}e^{-x/2}}{\Gamma(4)2^4}=0.12388</math> | ||
+ | |||
+ | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR distributions] |
Revision as of 18:25, 11 July 2011
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 X~Gamma(k,θ), 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 moment-generating 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 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:
The figure below shows this result using SOCR distributions