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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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up vote 3 down vote accepted

$L^2$ 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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I think it should be scale parameter $2$; it is the rate parameter (i.e. the inverse of the scale parameter) that is $\frac 1 2$. At least, that is how I've always used these terms. Change "scale" to "rate" and everything is good though. – guy May 22 '12 at 2:08
There are (at least) two different conventions about the meaning of $X \sim \Gamma(t,\lambda)$ and two different Wikipedia pages on Gamma random variables use two different conventions. I will edit my answer to avoid using either scale or rate. – Dilip Sarwate May 22 '12 at 2:13
Very informative, thank you. – mathjacks May 22 '12 at 13:37

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