Student's t-distribution
Compound distribution of normal with inverse gamma distribution

The location-scale t distribution results from compounding a Gaussian distribution (normal distribution) with mean μ {\displaystyle \ \mu \ } and unknown variance, with an inverse gamma distribution placed over the variance with parameters a = ν 2 {\displaystyle \ a={\frac {\ \nu \ }{2}}\ } and b = ν τ 2 2. {\displaystyle b={\frac {\ \nu \ \tau ^{2}\ }{2}}~.} 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 ν {\displaystyle \nu } and τ 2. {\displaystyle \ \tau ^{2}~.} The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. ν = 2 a, τ 2 = b a. {\displaystyle \ \nu =2\ a,\;{\tau }^{2}={\frac {\ b\ }{a}}~.}

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 t distribution arises naturally in many Bayesian inference problems.
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