Statistics 简明教程

Statistics - Gamma Distribution

伽马分布表示两参数族的连续概率分布。伽马分布通常由三种参数组合设计。

The gamma distribution represents continuous probability distributions of two-parameter family. Gamma distributions are devised with generally three kind of parameter combinations.

A shape parameter $ k $ and a scale parameter $ \theta $.
A shape parameter $ \alpha = k $ and an inverse scale parameter $ \beta = \frac{1}{ \theta} $, called as rate parameter.
A shape parameter $ k $ and a mean parameter $ \mu = \frac{k}{\beta} $.

每个参数都是正实数。Gamma 分配是由以下标准驱动的最大熵概率分配。

Each parameter is a positive real numbers. The gamma distribution is the maximum entropy probability distribution driven by following criteria.

Formula

其中——

Where −

${X}$ = Random variable.
${\psi}$ = digamma function.

Characterization using shape $ \alpha $ and rate $ \beta $

Probability density function

Gamma 分配的概率密度函数表示为：

Probability density function of Gamma distribution is given as:

Formula

其中——

Where −

${\alpha}$ = location parameter.
${\beta}$ = scale parameter.
${x}$ = random variable.

Cumulative distribution function

Gamma 分配的累积分布函数表示为：

Cumulative distribution function of Gamma distribution is given as:

Formula

其中——

Where −

${\alpha}$ = location parameter.
${\beta}$ = scale parameter.
${x}$ = random variable.
${\gamma(\alpha, \beta x)} $ = lower incomplete gamma function.

Characterization using shape $ k $ and scale $ \theta $

Probability density function

Gamma 分配的概率密度函数表示为：

Probability density function of Gamma distribution is given as:

Formula

其中——

Where −

${k}$ = shape parameter.
${\theta}$ = scale parameter.
${x}$ = random variable.
${\Gamma(k)}$ = gamma function evaluated at k.

Cumulative distribution function

Gamma 分配的累积分布函数表示为：

Cumulative distribution function of Gamma distribution is given as:

Formula

其中——

Where −

${k}$ = shape parameter.
${\theta}$ = scale parameter.
${x}$ = random variable.
${\gamma(k, \frac{x}{\theta})} $ = lower incomplete gamma function.