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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.
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A shape parameter $ k $ and a scale parameter $ theta $.
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A shape parameter $ alpha = k $ and an inverse scale parameter $ beta = frac{1}{ theta} $, called as rate parameter.
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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 −
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${X}$ = Random variable.
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${psi}$ = digamma function.
Characterization using shape $ alpha $ and rate $ beta $
Probability density function
Probability density function of Gamma distribution is given as:
Formula
Where −
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${alpha}$ = location parameter.
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${beta}$ = scale parameter.
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${x}$ = 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 −
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${alpha}$ = location parameter.
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${beta}$ = scale parameter.
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${x}$ = random variable.
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${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
Where −
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${k}$ = shape parameter.
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${theta}$ = scale parameter.
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${x}$ = random variable.
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${Gamma(k)}$ = 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 −
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${k}$ = shape parameter.
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${theta}$ = scale parameter.
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${x}$ = random variable.
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${gamma(k, frac{x}{theta})} $ = lower incomplete gamma function.
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