Statistics - Gamma Distribution

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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 $\mu = \frac{k}{\beta}$ .

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

Formula

${E[X] = k \theta = \frac{\alpha}{\beta} \gt 0 \ and \ is \ fixed. \\[7pt] E[ln(X)] = \psi (k) + ln( \theta) = \psi( \alpha) - ln( \beta) \ and \ is \ fixed. }$

Where −

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

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

Probability density function

Probability density function of Gamma distribution is given as:

Formula

${ f(x; \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1 } e^{-x \beta}}{\Gamma(\alpha)} \ where \ x \ge 0 \ and \ \alpha, \beta \gt 0 }$

Where −

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

Cumulative distribution function

Cumulative distribution function of Gamma distribution is given as:

Formula

${ F(x; \alpha, \beta) = \int_0^x f(u; \alpha, \beta) du = \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}}$

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

Probability density function of Gamma distribution is given as:

Formula

${ f(x; k, \theta) = \frac{x^{k - 1 } e^{-\frac{x}{\theta}}}{\theta^k \Gamma(k)} \ where \ x \gt 0 \ and \ k, \theta \gt 0 }$

Where −

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

Cumulative distribution function

Cumulative distribution function of Gamma distribution is given as:

Formula

${ F(x; k, \theta) = \int_0^x f(u; k, \theta) du = \frac{\gamma(k, \frac{x}{\theta})}{\Gamma(k)}}$

Where −

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

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Statistics - Gamma Distribution

Formula

Characterization using shape α \alpha and rate β \beta

Probability density function

Formula

Cumulative distribution function

Formula

Characterization using shape k k and scale θ \theta

Probability density function

Formula

Cumulative distribution function

Formula

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

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