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Gamma Distribution
Statistics - Gamma Distribution
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} $.
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.