Help on the density of Y/X, where X,Y~U(0,1)

[gimmytang]

Hi,
There are two i.i.d uniform random variables X and Y. Now I need to know the density of Y/X. My method is like this:
Let U=Y/X, V=X. Then the marginal density of U is what I need.

$$f_{U}(u)={\int_{-\infty}^{\infty}f_{U,V}(u,v)dv}={\int_{0}^{1}f_{X,Y}(u,uv)|v|dv}={\int_{0}^{1}vdv}=1/2$$
Now the question is that my result 1/2 is not a reasonable density since it's not integrated to 1. Can anyone point out where I am wrong?
gim

 

[uart]

I'm really rusty on this stuff, too rusty too even formalize a proper answer without brushing up on notation etc. So excuse me if this answer is a little vague but I think I know roughly what your problem is.

When you integrate out one of the variables of the joint distribution the limit is not just a simple "1", its actually function of the other variable. It helps if you try to sketch the joint distribution, the way I'm picturing it you should end up with something like,

f(u) = int(v,dv,0..1) : u in [0..1)
and
f(u) = int(v,dv,0..1/u) : u in [1..infinity)

This gives

f(u) = 0.5 : u in [0..1)
and
f(u)=0.5 u^(-2) : u in [1..infinity)

I'm not 100% sure it's correct but it looks reasonable

 

[gimmytang]

Thank you for your answer. It's correct since the distribution function is integrated to 1. I always have some confusion about the right domain of the variable since it's a function of the other variable.
gim