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Gamma Distribution (Gamma₍α,β₎) ○◂｜Definition｜1st｜20260604231540-00-⌔

Gamma distribution - Wikipedia

Gamma distribution

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In probability theory and statistics, the gamma distribution is a versatile two- parameter family of continuous probability distributions.¹ The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution.² There are two equivalent parameterizations in common use:

• With a shape parameter $\alpha$ and a scale parameter θ
• With a shape parameter $\alpha$ and a rate parameter ⁠ $\beta =1/\theta$ ⁠

In each of these forms, both parameters are positive real numbers.

The distribution has important applications in various fields, including econometrics, Bayesian statistics, and life testing.³ In econometrics, the (α, θ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an Erlang distribution for integer α values. Bayesian statisticians prefer the (α, β) parameterization, utilizing the gamma distribution as a conjugate prior for several inverse scale parameters, facilitating analytical tractability in posterior distribution computations.

The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a $1/x$ base measure) for a random variable X for which E [X] = αθ = α/β is fixed and greater than zero, and E [ln X] = ψ (α) + ln θ = ψ (α) − ln β is fixed (ψ is the digamma function).⁴

Printed 2026-06-28.

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Footnotes

1. “Gamma distribution | Probability, Statistics, Distribution | Britannica”. www.britannica.com. Archived from the original on 2024-05-19. Retrieved 2024-10-09. ↩

2. Weisstein, Eric W. “Gamma Distribution”. mathworld.wolfram.com. Archived from the original on 2024-05-28. Retrieved 2024-10-09. ↩

3. “Gamma Distribution | Gamma Function | Properties | PDF”. www.probabilitycourse.com. Archived from the original on 2024-06-13. Retrieved 2024-10-09. ↩

4. Park, Sung Y.; Bera, Anil K. (2009). “Maximum entropy autoregressive conditional heteroskedasticity model” (PDF). Journal of Econometrics. 150 (2): 219–230. Bibcode:2009JEcon.150..219P. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02. ↩

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