AP Statistics Curriculum 2007 Gamma
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- | === Gamma Distribution === | + | ==[[AP_Statistics_Curriculum_2007 | 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. | '''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. | ||
- | <br />'''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 <font size="3"><math>\lambda</math></font> is |
:<math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math> | :<math>P(x)=\frac{\lambda(\lambda x)^{h-1}}{(h-1)!}{e^{-\lambda x}}</math> | ||
- | For X | + | For <math>X\sim \operatorname{Gamma}(k,\theta)\!</math>, where <font size="3"><math>k=h</math></font> and <font size="3"><math>\theta=1/\lambda</math></font>, the gamma probability density function is given by |
:<math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math> | :<math>\frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}</math> | ||
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*e is the natural number (e = 2.71828…) | *e is the natural number (e = 2.71828…) | ||
*k is the number of occurrences of an event | *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 | + | *if k is a positive integer, then <font size="3"><math>\Gamma(k)=(k-1)!</math></font> 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. | + | *<font size="3"><math>\theta=1/\lambda</math></font> is the mean number of events per time unit, where <font size="3"><math>\lambda</math></font> 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 <font size="3"><math>\theta</math>=1/2=0.5</font>. If we want to find the mean number of calls in 5 hours, it would be <font size="3">5 <math>\times</math> 1/2=2.5</font>. |
*x is a random variable | *x is a random variable | ||
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where | where | ||
- | *if k is a positive integer, then <math>\Gamma(k)=(k-1)!</math> is the gamma function | + | *if k is a positive integer, then <font size="3"><math>\Gamma(k)=(k-1)!</math></font> is the gamma function |
- | *<math>\gamma(k,x/\theta)=\int_0^{x/\theta}t^{k-1}e^{-t}dt</math> | + | *<math>\textstyle\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 | <br />'''Moment generating function''': The gamma moment-generating function is | ||
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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. | 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=== | ===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. | 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: | + | One fish every 1/2 hour means we would expect to get <font size="3"><math>\theta=1 / 0.5=2</math></font> fish every hour on average. Using <font size="3"><math>\theta=2</math></font> and <font size="3"><math>k=4</math></font>, 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> | :<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] | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR distributions] | ||
<center>[[Image:Gamma.jpg|600px]]</center> | <center>[[Image:Gamma.jpg|600px]]</center> | ||
+ | |||
+ | |||
+ | ===Normal Approximation to Gamma distribution=== | ||
+ | |||
+ | Note that if \( \{X_1,X_2,X_3,\cdots \}\) is a sequence of independent [[AP_Statistics_Curriculum_2007_Exponential|Exponential(b) random variables]] then \(Y_k = \sum_{i=1}^k{X_i} \) is a [http://www.math.uah.edu/stat/special/Gamma.html random variable with gamma distribution] with the following shape parameter, '''k''' (positive integer indicating the number of exponential variable in the sum) and scale parameter '''b''' (which is the exponential parameter). By the [[AP_Statistics_Curriculum_2007_Limits_CLT|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 <math>k\longrightarrow \infty</math>. | ||
+ | |||
+ | For the example above, \(\Gamma(k=4, \theta=2)\), the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html SOCR Normal Distribution Calculator] can be used to obtain an estimate of the area of interest as shown on the image below. | ||
+ | |||
+ | <center>[[Image:EBook_Gamma_Fig2.png|500px]]</center> | ||
+ | |||
+ | The probabilities of the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html real Gamma] and [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html approximate Normal] distributions (on the range [2:4]) are not identical but are sufficiently close. | ||
+ | |||
+ | <center> | ||
+ | {| class="wikitable" style="text-align:center; width:75%" border="1" | ||
+ | |- | ||
+ | ! Summary|| [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html \(\Gamma(k=4, \theta=2)\) ] || [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html \(Normal(\mu=8, \sigma^2=4)\) ] | ||
+ | |- | ||
+ | | 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 | ||
+ | |- | ||
+ | ! colspan=3|Probability Areas | ||
+ | |- | ||
+ | | <2|| 0.018988|| 0.066807 | ||
+ | |- | ||
+ | | [2:4]|| 0.123888||0.091848 | ||
+ | |- | ||
+ | | >4|| 0.857123||0.841345 | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <hr> | ||
+ | * SOCR Home page: http://www.socr.ucla.edu | ||
+ | |||
+ | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Gamma}} |
Current revision as of 17:34, 23 June 2012
Contents |
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
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 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
![](/socr/uploads/thumb/1/1b/Gamma.jpg/600px-Gamma.jpg)
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 the following shape parameter, k (positive integer indicating the number of exponential variable in the sum) and scale parameter b (which is the exponential parameter). 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 .
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.
![](/socr/uploads/thumb/e/ee/EBook_Gamma_Fig2.png/500px-EBook_Gamma_Fig2.png)
The probabilities of the real Gamma and approximate Normal distributions (on the range [2:4]) are not identical but are sufficiently close.
Summary | \(\Gamma(k=4, \theta=2)\) | \(Normal(\mu=8, \sigma^2=4)\) |
---|---|---|
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 |
Probability Areas | ||
<2 | 0.018988 | 0.066807 |
[2:4] | 0.123888 | 0.091848 |
>4 | 0.857123 | 0.841345 |
- SOCR Home page: http://www.socr.ucla.edu
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