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Let $U=X/Y$ and $V=XY$, which means $X=\sqrt{UV}$ and $Y=\sqrt{V/U}$.

Now I am doing this for an assignment and I do not want to get given answers.

I need to understand what the above interaction looks like geometrically.

I want to know how I can go about determining joint pdf and typical probability.

Again, just wanting some pointers.

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You might try to compute the joint ditribution, as $F_{U,V}(u,v)=P(U\le u,V\le v)=P(X\le u Y,X Y \le v)$. This could be looked at geometrically (find the area of the corresponding event inside the unit square).

Update: because $X,Y$ are iid and uniform over $[0,1]$, the probability of the event of interest ($Y \ge X/u \cap Y \le v/X$) can be computed simply calculating the relevant area.

Depending on the ranges of $u,v$, there are four cases, which I illustrate below (the fourth is when $v>1$, which is simpler).

enter image description here

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  • $\begingroup$ I am finding it hard to think about how I go from say the pdf of a random variable, X ~U[0,1] which has pdf of 1/(1-0) = 1 to something like U=X/Y. So I am not sure how exactly I move on $\endgroup$
    – Jonathan Miller
    Mar 30, 2015 at 12:08
  • $\begingroup$ There is no easy general way to visualize the density of X (and Y) to $X/Y$. My answer gives you a way to compute it. If you need more detail, tell me. $\endgroup$
    – leonbloy
    Mar 30, 2015 at 16:27
  • $\begingroup$ I need more detail. I understand I should integrate the joint pdf of the uniform rectangle, which i believe to be 1. If I am right, I do not know what bounds I should use. $\endgroup$
    – Jonathan Miller
    Mar 30, 2015 at 16:39

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