# Thread: Change of variables when the function is not 1-1

1. ## Change of variables when the function is not 1-1

I ran into this problem:
Suppose $\displaystyle X, Y IID N(0,1)$, obtain the pdf for $\displaystyle U=\frac{X}{\sqrt{X^2+Y^2}}$.
Because of the nasty square root, I can't seem to find the appropriate transformation functions for a change of variables. I tried letting $\displaystyle R=\sqrt{X^2+Y^2}$ and looking for the joint pdf of $\displaystyle U$ and $\displaystyle R$, but the function is not 1-1. Since $\displaystyle Z=\frac{Y}{X}$ is a standard cauchy, I've also tried breaking $\displaystyle U$ into: $\displaystyle -\frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x<0, 0$ for $\displaystyle x=0, \frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x>0$, but that doesn't help much.
What am I supposed to do when I see this type of questions? I need some hints.

2. Hello,

I'm giving a quick thought. Can't you first find the pdf of $\displaystyle U^2$ (by finding the joint pdf of $\displaystyle U^2$ and $\displaystyle R^2$) ? Don't forget changing the boundaries, as you have symmetry for x and for y.
I don't know what method you use, so please show a little work and I may be more able to help you !

3. Originally Posted by Crescent
I ran into this problem:
Suppose $\displaystyle X, Y IID N(0,1)$, obtain the pdf for $\displaystyle U=\frac{X}{\sqrt{X^2+Y^2}}$.
Because of the nasty square root, I can't seem to find the appropriate transformation functions for a change of variables. I tried letting $\displaystyle R=\sqrt{X^2+Y^2}$ and looking for the joint pdf of $\displaystyle U$ and $\displaystyle R$, but the function is not 1-1. Since $\displaystyle Z=\frac{Y}{X}$ is a standard cauchy, I've also tried breaking $\displaystyle U$ into: $\displaystyle -\frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x<0, 0$ for $\displaystyle x=0, \frac{1}{\sqrt{1+Z^2}}$ for $\displaystyle x>0$, but that doesn't help much.
What am I supposed to do when I see this type of questions? I need some hints.
$\displaystyle $$U is obviously the projection onto the x-axis of a uniform RV on the unit circle. That is \displaystyle$$ U$ has the same distribution as $\displaystyle X=\cos(A),\ A\sim U(0,2\pi)$

Can you take it from there?

CB