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} $.

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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