# Infinite boundary for random variables

I have a question

Suppose that X and Y are random variables with joint pdf is given by

and zero otherwise.

I need to find marginal and conditional pdf's.But I don't know how to intagrate over an infinite boundary in order to find the marginal pdf's.Can someone explain it to me?.Thank you.

The joint density $f_{X,Y}$ is defined on $\mathbb R^2$ by $$f_{X,Y}(x,y)=\frac1{2x^2y}\mathbf 1_{x\gt1}\mathbf 1_{1/x\lt y\lt x}=\frac1{2x^2y}\mathbf 1_{y\gt0}\mathbf 1_{x\gt\max(y,1/y)},$$ hence $$f_X(x)=\mathbf 1_{x\gt1}\int_{1/x}^x\frac1{2x^2y}\mathrm dy,\qquad f_Y(y)=\mathbf 1_{y\gt0}\int_{\max(y,1/y)}^\infty\frac1{2x^2y}\mathrm dx.$$ Surely you can continue.
• For every property $P$, $\mathbf 1_P=1$ is $P$ holds and $\mathbf 1_P=0$ otherwise. Manipulation: the goal is to write properly the domain where $f_{X,Y}$ is not zero, going at it slowly shows the task is not that difficult. Re integrals on infinite intervals: sorry but you will have to get used to that if you continue studying probability since densities defined on "infinite" domains are quite banal (but you do not say what the problem is). – Did Oct 20 '14 at 16:51