Editing Student's t-distribution (section)

====Compound distribution of normal with inverse gamma distribution====
The location-scale {{mvar|t}}&nbsp;distribution results from [[compound distribution|compounding]] a [[Normal distribution|Gaussian distribution]] (normal distribution) with [[mean]] <math>\ \mu\ </math> and unknown [[variance]], with an [[inverse gamma distribution]] placed over the variance with parameters <math>\ a = \frac{\ \nu\ }{ 2 }\ </math> and <math>b = \frac{\ \nu\ \tau^2\ }{ 2 } ~.</math> In other words, the [[random variable]] ''X'' is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is [[marginalized out]] (integrated out).

Equivalently, this distribution results from compounding a Gaussian distribution with a [[scaled-inverse-chi-squared distribution]] with parameters <math>\nu</math> and <math>\ \tau^2 ~.</math> The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. <math>\ \nu = 2\ a, \; {\tau}^2 = \frac{\ b\ }{ a } ~.</math>

The reason for the usefulness of this characterization is that in [[Bayesian statistics]] the inverse gamma distribution is the [[conjugate prior]] distribution of the variance of a Gaussian distribution. As a result, the location-scale {{mvar|t}}&nbsp;distribution arises naturally in many Bayesian inference problems.<ref>{{Cite book |title=Bayesian Data Analysis |vauthors=Gelman AB, Carlin JS, Rubin DB, Stern HS |publisher=Chapman & Hal l|year=1997 |isbn=9780412039911 |edition=2nd |location=Boca Raton, FL |pages=68 }}</ref>