Can someone provide a description where the Jacobian would be used in probability for variable transformation? It seems to me like all the transformations that I have done in the text book  can be dealt with the Distribution function of the transform method. I was wondering if someone would be able to create an example that would showcase the use of the Jacobian matrix and hopefully help me understand intuitively what is going on?
 A: Suppose $X$ and $Y$ are independent random variables with Gamma distributions
$$
\frac 1 {\Gamma(\alpha)} \left( \frac x \lambda \right)^\alpha e^{-x/\lambda} \ \frac{dx} x \text{ on }(0,\infty)\quad\text{and}\quad \frac 1 {\Gamma(\beta)} \left( \frac y \lambda \right)^\beta e^{-y/\lambda} \ \frac{dy} y \text{ on }(0,\infty).
$$
Let
$$
\begin{cases} U = X + Y, \\[10pt]  V = \dfrac X {X+Y}. \end{cases} 
$$
Show that


*

*$U$ has a Gamma distribution $$ \frac 1 {\Gamma(\alpha+\beta)} \left( \frac u \lambda \right)^{\alpha+\beta} e^{-u/\lambda}\ \frac{du} u \quad\text{on }(0,\infty), $$ and

*$V$ has a Beta distribution $$ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} v^{\alpha-1} (1-v)^{\beta-1} \, dv \text{ on } (0,1),  $$ and

*$U,V$ are independent.


Another example is the derivation of the density of the usual $F$-distribution, showing it's (if I recall correctly!) the square root of a rational function.  Remember that it is
$$
\frac{\chi^2_n/n}{\chi^2_m/m}
$$
where the two chi-square random variables are independent.
