# Finding joint support of $(XY,X/Y)$ where $(X,Y)$ has joint pdf $1/x^2y^2$ for $x,y\ge1$

$$X$$ and $$Y$$ are random variables with joint pdf $$\frac{1}{x^2y^2}\qquad,\, x\ge1,y\ge1$$

Set $$U=XY, V=X/Y$$

Explain why the joint range of $$U$$ and $$V$$ is given by:

$$\{(u,v):v\in(0,1),u\ge1/v\} \cup \{(u,v):v∈[1,\infty),u\ge v\}$$

I don't understand what this question wants. Can anyone explain to me how to approach this? In response to another bit of the same question, I have successfully shown that the joint pdf of $$U$$ and $$V$$ is $$\frac{1}{2u^2v}$$ by going through and subbing in the new variables and obtaining the Jacobian, and then gone on to obtain the marginal of $$V$$ (I think $$1/2v$$) but this aspect of the question in relation to range is stumping me.

Any general advice (given in simple terms!) would be welcomed.

PS the question also asks for a sketch.

• The joint support of $(U,V)$ is precisely what's needed to write the joint density of $(U,V)$ in the first place. – StubbornAtom Dec 1 '18 at 22:03
• You might want to add the self-study tag. – StubbornAtom Dec 2 '18 at 12:14
• Done and thank you all for the help. It was very useful. – StatisticsPersonInTraining Dec 2 '18 at 12:17

Find the support of X and Y joint distribution (which is $$(x\ge 1) \cap (y \ge 1)$$).
Convert the points in X-Y support into U-V plain by the relation $$U=XY$$ and $$V=X/Y$$, you get the support of U-V joint distribution. For example, (X,Y) = (1,1) ==> (U,V) = (1,1), (X,Y) = (2,2) ==> (U,V) = (4,1).
In your situation, there are two lines $$(1,1)$$ to $$(1,\infty)$$ and $$(1,1)$$ to $$(\infty,1)$$ in X-Y graph, and you need to convert these two lines from X-Y graph to U-V graph, then you get the answer.