Gamma Distribution

Probability Density Function

A random variable $X$ has a gamma distribution $X \sim G a m m a (α, β)$ if and only if it has the pdf

f (x; α, β) = \frac{1}{β^{α} Γ (α)} x^{α - 1} e^{- x / β} for x > 0

where $α > 0$ and $B > 0$ and $Γ (α)$ is the Gamma function

\begin{aligned} \int_{- \infty}^{\infty} f (x) d x & = \int_{0}^{\infty} \frac{1}{β^{α} Γ (α)} x^{α - 1} e^{- x / β} d x \\ = \frac{1}{β^{α} Γ (α)} \int_{0}^{\infty} x^{α - 1} e^{- x / β} d x (y = \frac{x}{β}) \\ = \frac{1}{β^{α} Γ (α)} \int_{0}^{\infty} (β y)^{α - 1} e^{- y} β d y (d x = β d y) \\ = \frac{1}{β^{α} Γ (α)} β^{α} \int_{0}^{\infty} y^{α - 1} e^{- y} d y \\ = \frac{1}{Γ (α)} Γ (α) \\ = 1 \end{aligned}

Mean

\begin{aligned} μ = E (X) & = \int_{0}^{\infty} x \frac{1}{β^{α} Γ (α)} x^{α - 1} e^{- x / β} d x \\ = \frac{1}{β^{α} Γ (α)} \int_{0}^{\infty} x^{α + 1 - 1} e^{- x / β} \\ = \frac{1}{β^{α} Γ (α)} β^{α + 1} Γ (α + 1) \int_{0}^{\infty} \frac{1}{β^{α + 1} Γ (α + 1)} x^{α + 1 - 1} e^{- x / β} d x \\ = β \frac{Γ (α + 1)}{Γ (α)} \cdot 1 (PDF of G a m m a (α + 1, β)) \\ = \frac{β α Γ (α)}{Γ (α)} \\ μ & = β α \end{aligned}

Second moment

\begin{aligned} E (X^{2}) & = \int_{0}^{\infty} x^{2} \frac{1}{β^{α} Γ (α)} x^{α - 1} e^{- x / β} d x \\ = \frac{d^{2}}{d t^{2}} M_{X} (t) \\ = \frac{d}{d t} α β (1 - t β)^{- α - 1} \\ = α β (- α - 1) (1 - t β)^{- α - 2} (- β) |_{t = 0} \\ = - β^{2} α (- (α + 1)) (1 - 0)^{- α - 2} \\ = β^{2} α (α + 1) \end{aligned}

Variance

\begin{aligned} V a r (X) & = E (X^{2}) - [E (X)]^{2} \\ = β^{2} α (α + 1) - (α β)^{2} \\ = β^{2} (α (α + 1) - α^{2}) \\ = β^{2} (α (α + 1 - α)) \\ = β^{2} (α (1)) \\ V a r (X) & = β^{2} α \end{aligned}

Exponential Distribution is a special case of the gamma distribution when $α = 1$

The Chi-Squared Distribution is another special case of the gamma distribution with $α = \frac{v}{2}$ and $β = 2$

The rth moment of a $G a m m a (α, β)$ distribution is given by

E (X^{r}) = β^{r} \frac{Γ (α + r)}{Γ (α)}