Random Variables Transformation

[libelec]

Homework Statement

X, Y, Z random variables, independent of each other, with uniform distribution in (0,1). C = XY and R = Z². Without using the joint probability function, find P(C>R).

The Attempt at a Solution

So far:

P(C > R) = P(C - R > 0) = P(XY - Z² > 0) = P(g(X,Y,Z) > 0).

Now, I don't know how to find g^-1(-infinity, 0) = {(x,y,z) in (0,1)³ : xy - z² > 0} to continue to operate. Or any other way to solve this.

Any suggestions?

 

[JSuarez]

Try $-\sqrt{XY} < Z < \sqrt{XY}$ instead, and remember that all variables are uniform over (0,1).

 

[libelec]

Let me see.

P(C > R) = P(Z²<XY) = P($$- \sqrt {XY} < Z < \sqrt {XY}$$) = F_Z($$\sqrt {XY}$$) - F_Z(-$$\sqrt {XY}$$), where in the second equality I used that all values of XY are positive and in the last one that F_Z is absolutely continuous.

But now, since Z ~ U(0,1), its F_Z(z) = z 1{0<z<1} + 1{z>1}. So F_Z(-$$\sqrt {XY}$$) = 0, because F_Z(z) is 0 for all z<0.

So, I would get that P(C > R) = F_Z($$\sqrt {XY}$$) = sqrt(xy) 1{0<sqrt(xy)<1} + 1{sqrt(xy)>1}. XY is bounded by 1, so sqrt(xy) can't be >1. Then this is sqrt(xy) 1{0<sqrt(xy)<1}.

I'm sorry, but I don't know how to continue...

 

[libelec]

And now what do I do? How can I use that X and Y have uniform distributions?