# Joint Distribution, expected value correlation of a graphed triangle (Fixed)

• Dec 14th 2012, 11:24 AM
WUrunner
Joint Distribution, expected value correlation of a graphed triangle (Fixed)
Suppose X and Y are two independent random variables each distributed as $Uniform(0,1)$.

1) Find the joint distribution of X and Y.

2) Let $U = \cos(2 \pi Y)\sqrt{-2\ln(X)}$ and $V = \sin(2\pi Y)\sqrt{-2\ln(X)}$. Find the joint distribution
of U and V assuming that the transformation is one-to-one?

3) Find the marginal distributions of U and V?

For 1, I got that my distribution is 1 by multiplying the two distributions together. For 2, I have been getting stuck trying to get my equations in terms of X and Y to perform the transformation. I simplified and got $Y =$ $(\frac{1}{2})$ $\tan^{-1}(\frac{V}{U})$ but something seems wrong about this and am having trouble trying to get X. For 3, I am obviously stuck and even not totally sure on the support. Any help is appreciated.
• Dec 14th 2012, 01:11 PM
chiro
Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)
Hey WUrunner (and thanks for fixing the latex up).

For V you can use the transformation theorem in probability and for U you can write things in terms of V.

Also is sin(2) meant to be sin(2*pi*Y)? If not then the tangent expression will be wrong.
• Dec 14th 2012, 08:33 PM
WUrunner
Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)
Thanks for pointing out my mistake. I somehow missed my sin function
• Dec 14th 2012, 09:35 PM
chiro
Re: Joint Distribution, expected value correlation of a graphed triangle (Fixed)
Also you want to look at ratio distributions to get the distribution of a ratio of two independent variables:

Ratio distribution - Wikipedia, the free encyclopedia