Continuous random variable

Gamma distribution

A gamma distributed random variable $𝑌 \sim G a m m a (𝑎, 𝜆)$ where $𝑎, 𝜆 > 0$ is described by the probability density function #m/def/prob

𝑓 𝑌 (𝑦) = 𝜆 𝑎 Γ (𝑎) 𝑦 𝑎 - 1 e - 𝜆 y

where $Γ$ is the gamma function.

Properties

Expectation: $𝔼 [𝑌] = 𝑎 𝜆$
Variance: $V a r [𝑋] = 𝑎 𝜆 2$
Moments: $𝔼 [𝑌 𝑛] = 𝜆 - 𝑛 Γ (𝑎 + 𝑛) Γ (𝑎)$ for $𝑛 > - 𝑎$

Furthermore

The sum of $𝑎$ independent exponential random variables is $G a m m a (𝑎, 𝜆)$
Conjugate prior to Poisson #to/elaborate
A special case is the Chi-squared distribution

Relationship to other distributions

By the Central limits theorem for integer $𝑛$ , $G a m m a (𝑛, 𝜆) ⇝ N (𝑛 𝜆, 𝑛 𝜆 2)$ as $𝑛 \to \infty$ .

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