In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
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normal-inverse-gamma
Probability density function |
Parameters |
location (real)
(real)
(real)
(real) |
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Support |
![{\displaystyle x\in (-\infty ,\infty )\,\!,\;\sigma ^{2}\in (0,\infty )}](//wikimedia.org/api/rest_v1/media/math/render/svg/9752767a3db2a43c037cfefbe059a0409c416c18) |
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PDF |
![{\displaystyle {\frac {\sqrt {\lambda }}{\sqrt {2\pi \sigma ^{2}}}}{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d3d4040d8090127f083255889c6cfb8f3073cbc1) |
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Mean |
![{\displaystyle \operatorname {E} [x]=\mu }](//wikimedia.org/api/rest_v1/media/math/render/svg/d60f5921cca1c75d673eb70db395bf3a88f9170f)
, for ![{\displaystyle \alpha >1}](//wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd)
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Mode |
![{\displaystyle \sigma ^{2}={\frac {\beta }{\alpha +1+1/2}}\;{\textrm {(univariate)}},\sigma ^{2}={\frac {\beta }{\alpha +1+k/2}}\;{\textrm {(multivariate)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a2d8570284c452dabd6b54e3081c45cad00113e6) |
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Variance |
, for ![{\displaystyle \alpha >1}](//wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd)
, for ![{\displaystyle \alpha >2}](//wikimedia.org/api/rest_v1/media/math/render/svg/432334d220d6e1b0340cc2a37531d0327494a8e2)
, for ![{\displaystyle \alpha >1}](//wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd) |
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