Statistics Gamma Distribution – lesscss

By | October 18, 2019

Statistics – Gamma Distribution

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} $.

Gamma Distribution

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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