Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

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 $h^{-1}(l) = l^2$.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.