I have a question

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

enter image description here

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.

  • $\begingroup$ Thank you but I couldn't understand what you did here.Can you explain it more. $\endgroup$ – Zapdos Oct 20 '14 at 16:43
  • $\begingroup$ Sure--which part escapes you and why? Please be specific. $\endgroup$ – Did Oct 20 '14 at 16:44
  • $\begingroup$ I am really new at these concepts.Firstly,I have never seen the notation 1_x>1.And the second is,I couldn't understand how you manipulated these boundaries.And lastly,f_Y(y) includes an infinite boundary.I don't know how to integrate something like this.Thank you. $\endgroup$ – Zapdos Oct 20 '14 at 16:47
  • $\begingroup$ 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). $\endgroup$ – Did Oct 20 '14 at 16:51
  • $\begingroup$ Ok,so the problem is say I integrate this equation and the last step is putting boundaries to the result and subtracting each other,am I right.Lower boundary is max(y,1/y) but the upper boundary is "infinity".How can I insert infinity to a variable.Obviously there is something I don't know. $\endgroup$ – Zapdos Oct 20 '14 at 16:56

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