Inverse-gamma distribution

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Probability density function
Cumulative distribution function
Parameters shape (real)
scale (real)
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean for
Variance for
Skewness for
Excess kurtosis for
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.


Probability density function

The inverse gamma distribution's probability density function is defined over the support

with shape parameter and scale parameter .

Cumulative distribution function

The cumulative distribution function is the regularized gamma function

where the numerator is the upper incomplete gamma function and the denominator is the gamma function.

Related distributions

  • If and and then is an inverse-chi-square distribution
  • If , then is a Gamma distribution
  • A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.

Derivation from Gamma distribution

The pdf of the gamma distribution is

and define the transformation then the resulting transformation is

Replacing with ; with ; and with results in the inverse-gamma pdf shown above

See also

it:Variabile casuale Gamma inversa