Difference between revisions of "Gamma distribution"
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| − | + | = Gamma(alpha'',beta'') = | |
| + | |||
| + | Creates a gamma distribution with shape parameter alpha and scale | ||
| + | parameter beta. The scale parameter, beta, is optional and defaults to | ||
| + | beta=1. The gamma distribution is bounded below by zero (all sample | ||
| + | points are positive) and is unbounded from above. It has a | ||
| + | theoretical mean of A*B and a theoretical variance of alpha*beta^2. | ||
| + | When alpha>1, the distribution is unimodal with the mode at | ||
| + | (alpha-1)*beta. An exponential distribution results when alpha=1. As | ||
| + | alpha-->oo, the gamma distribution approaches a normal distribution | ||
| + | in shape. | ||
| + | |||
| + | The gamma distribution encodes the time required for alpha events to | ||
| + | occur in a Poisson process with mean arrival time of beta. | ||
| + | |||
| + | ===Note=== | ||
| + | Some textbooks use Rate=1/beta, instead of beta, as the scale parameter. | ||
| + | |||
| + | = When to use = | ||
| + | |||
| + | Use the gamma distribution with alpha>1 if you have a sharp lower | ||
| + | bound of zero but no sharp upper bound, a single mode, and a | ||
| + | positive skew. The [[LogNormal]] distribution is also an option in this | ||
| + | case. Gamma() is especially appropriate when encoding arrival | ||
| + | times for sets of events. A gamma distribution with a large value | ||
| + | for alpha is also useful when you wish to use a bell-shaped curve for | ||
| + | a positive-only quantity. | ||
| + | |||
| + | = Library = | ||
| + | |||
| + | Distribution | ||
| + | |||
| + | = See Also = | ||
| + | |||
| + | * [[Dens_Gamma]] - probability density at x | ||
| + | * [[GammaI]] -- cumulative density at x, incomplete gamma function | ||
| + | * [[GammaIInv]] -- inverse cumulative density | ||
| + | * [[GammaFn]] -- the gamma function | ||
| + | * [[Beta]], [[Exponential]], [[LogNormal]] -- related distributions | ||
Revision as of 20:14, 3 October 2007
Gamma(alpha,beta)
Creates a gamma distribution with shape parameter alpha and scale parameter beta. The scale parameter, beta, is optional and defaults to beta=1. The gamma distribution is bounded below by zero (all sample points are positive) and is unbounded from above. It has a theoretical mean of A*B and a theoretical variance of alpha*beta^2. When alpha>1, the distribution is unimodal with the mode at (alpha-1)*beta. An exponential distribution results when alpha=1. As alpha-->oo, the gamma distribution approaches a normal distribution in shape.
The gamma distribution encodes the time required for alpha events to occur in a Poisson process with mean arrival time of beta.
Note
Some textbooks use Rate=1/beta, instead of beta, as the scale parameter.
When to use
Use the gamma distribution with alpha>1 if you have a sharp lower bound of zero but no sharp upper bound, a single mode, and a positive skew. The LogNormal distribution is also an option in this case. Gamma() is especially appropriate when encoding arrival times for sets of events. A gamma distribution with a large value for alpha is also useful when you wish to use a bell-shaped curve for a positive-only quantity.
Library
Distribution
See Also
- Dens_Gamma - probability density at x
- GammaI -- cumulative density at x, incomplete gamma function
- GammaIInv -- inverse cumulative density
- GammaFn -- the gamma function
- Beta, Exponential, LogNormal -- related distributions
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