Definition:Gamma Distribution
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Definition
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Img X = \R_{\ge 0}$.
$X$ is said to have a Gamma distribution if it has probability density function:
- $\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
for $\alpha, \beta > 0$, where $\Gamma$ is the Gamma function.
This is written:
- $X \sim \map \Gamma {\alpha, \beta}$
The validity of the material on this page is questionable. In particular: In my Oxford dictionary, the parameters are the other way round. They present it as $\map \Gamma {\lambda, r}$ where $\lambda$ takes the role of $\beta$ and $r$ takes the role of $\alpha$. Hence that suggests the presentation should be $\map \Gamma {\beta, \alpha}$. Can this be reviewed? The original page on this was unsourced. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also defined as
The gamma distribution can also be seen in the form:
- $\map f x = \dfrac {x^{\alpha - 1} e^{-\alpha / \beta} } {\beta^\alpha \map \Gamma \alpha}$
Also see
- Results about the Gamma distribution can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions
- Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GammaDistribution.html