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Below is an excerpt from a paper. I would like to understand how equation 1.4 was derived. This is not the PDF on the wikipedia page or my reference book for the multivariate student t distribution or chi square distribution. $C$ is a constant.

Additionally, what's the significance of the PDF for $L$ vs $L^2$?

Student t

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$L^2$ is a Gamma random variable with mean $m$ and order parameter $m/2$, and thus its density is of the form $g(x) = Bx^{m/2-1}\exp(-x/2), ~~ x \geq 0$
where $B$ is a constant. Hence, $L$ has density $f(l) = 2lg(l^2) = Cl^{m-1}\exp(-l^2/2)$ where the transformation from $g(x)$ to $f(l)$ uses the standard formula (see almost any text on probability theory) $f(l) = g(h^{-1}(l))\left|\frac{\mathrm dh^{-1}(l)}{\mathrm dl}\right|$ with $h^{-1}(\cdot)$ being the inverse of the map $h$ that transforms $L^2$ into $L$. In other words, $h(y) = \sqrt{y}$ and $h^{-1}(l) = l^2$.

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    Very informative, thank you.2012-05-22