Statistics/Distributions/Gamma

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The Gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions.

The probability distribution function is:

f_x (x) = \begin{cases} \frac{1}{a^p \Gamma (p)} x^{p-1} e^{-x/a}, & \mbox{if } x \ge 0 \\ 0, & \mbox{if } x < 0 \end{cases}

The cumulative distribution function cannot be found unless p=1, in which case the Gamma distribution becomes the exponential distribution. The Gamma distribution of the stochastic variable X is denoted as X \in \Gamma (p,a) .

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1 / θ, called a rate parameter:

g(x;\alpha,\beta) = K x^{\alpha-1} e^{-\beta\,x} \ \mathrm{for}\ x > 0 \,\!.

where the K constant can be calculated setting the integral of the density function as 1:

\int_{-\infty}^{+\infty}g(x;\alpha,\beta) \mathrm{d}t \, = \int_{0}^{+\infty} K x^{\alpha-1} e^{-\beta\,x} \mathrm{d}x \, = 1

following:

K \int_{0}^{+\infty} x^{\alpha-1} e^{-\beta\,x} \mathrm{d}x \, = 1
K = \frac{1}{\int_{0}^{+\infty} x^{\alpha-1} e^{-\beta\,x} \mathrm{d}x}

and, with change of variable y = βx :

\begin{align} K &= \frac{1}{\int_{0}^{+\infty} \frac{y^{\alpha-1}}{\beta^{\alpha - 1}} e^{-y} \frac{\mathrm{d}y}{\beta}} \\ &= \frac{1}{\frac{1}{\beta^{\alpha}}\int_{0}^{+\infty} y^{\alpha-1} e^{-y} \mathrm{d}y} \\ &= \frac{\beta^{\alpha}}{\int_{0}^{+\infty} y^{\alpha-1} e^{-y} \mathrm{d}y} \\ &= \frac{\beta^{\alpha}}{\Gamma(\alpha)} \end{align}

following:

g(x;\alpha,\beta) = x^{\alpha-1} \frac{\beta^{\alpha} \, e^{-\beta\,x} }{\Gamma(\alpha)} \ \mathrm{for}\ x > 0 \,\!.

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