# Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base.

I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then two variables that are functions of those, Y=$\frac{X_1}{X_2}$ and Z=${X_1}{X_2}$. I need to find the joint density of Y and Z on their support.

Solving for $X_1$ and $X_2$ in term of Y and Z gives me $X_1$=$\sqrt{YZ}$ and $X_2$=$\sqrt{Z/Y}$. Solving for the Jacobian, I get $\frac{\sqrt{Z/Y}}{2\sqrt{YZ}}$ (simplified from $\frac{Y \sqrt{\frac{Z}{Y}}}{4 Y \sqrt{Y Z}}+\frac{Z \sqrt{\frac{Z}{Y}}}{4 Z \sqrt{Y Z}}$). I know f($X_1,Y_1$)=1 over their support, and it would be nice if f(Y,Z) were just 1 times the Jacobian, but I think I need to address the support of Y and Z?

Thanks.

• Hint : You have $5\le X_1\le 10$ then $5\le \sqrt{YZ}\le 10$. Similarly with the another one. After that plot the region. – Tunk-Fey Apr 17 '14 at 6:32
• @Tunk-Fey, thanks for letting me know I'm on the right track. I guess I'm not clear how the constraints come in for the joint density. I know that once I have f(Y,Z), I would need the limits of integration to find a marginal density, but from my notes, it looks like f(Y,Z)=f($X_1,X_2$) x Jacobian. Am I plotting this on the Y-Z grid? – user143719 Apr 17 '14 at 13:00
• Plotting this on Y-Z, I have a max Z of Z=100Y for .5<Y<1 and Z=100/Y for 1<Y<2.5, and a min Z of Z=25/Y for .5<Y<1.25 and Z=16Y for 1.25<Y<2.5. Now, the page only has one field to fill in, and it asks for the density "on their support," so I suspect I don't need to list the support, just the function. And isn't that just the original density times the Jacobian? Office hours are about to start, so maybe I'll have an answer soon. – user143719 Apr 17 '14 at 14:06

Actually, I don't need the support of Y and Z at all. "f($X_1,X_2$)=1 over their support" means that f($X_1,X_2$)=$\frac{1}{(10-4)(1-5)}$=$\frac{1}{30}$, so f(Y,Z)=$\frac{\sqrt{Z/Y}}{2\sqrt{YZ}}$*$\frac{1}{30}$=$\frac{\sqrt{Z/Y}}{60\sqrt{YZ}}$