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Gamma Distribution
  • 时间:2024-11-03

Statistics - Gamma Distribution


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

    A shape parameter $ k $ and a scale parameter $ heta $.

    A shape parameter $ alpha = k $ and an inverse scale parameter $ eta = frac{1}{ heta} $, called as rate parameter.

    A shape parameter $ k $ and a mean parameter $ mu = frac{k}{eta} $.

Gamma Distribution

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

Formula

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

Where −

    ${X}$ = Random variable.

    ${psi}$ = digamma function.

Characterization using shape $ alpha $ and rate $ eta $

Probabipty density function

Probabipty density function of Gamma distribution is given as:

Formula

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

Where −

    ${alpha}$ = location parameter.

    ${eta}$ = scale parameter.

    ${x}$ = random variable.

Cumulative distribution function

Cumulative distribution function of Gamma distribution is given as:

Formula

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

Where −

    ${alpha}$ = location parameter.

    ${eta}$ = scale parameter.

    ${x}$ = random variable.

    ${gamma(alpha, eta x)} $ = lower incomplete gamma function.

Characterization using shape $ k $ and scale $ heta $

Probabipty density function

Probabipty density function of Gamma distribution is given as:

Formula

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

Where −

    ${k}$ = shape parameter.

    ${ heta}$ = 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

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

Where −

    ${k}$ = shape parameter.

    ${ heta}$ = scale parameter.

    ${x}$ = random variable.

    ${gamma(k, frac{x}{ heta})} $ = lower incomplete gamma function.

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