I have the following question, for which the first part of (a), I had no trouble with, however, when it comes to sketching the region described (highlighted in yellow) - problem is in the link further below. I'm a bit lost on how to even approach this.
for context and to know that I haven't just run here, I have watched several videos which I thought would help on sketching the range. However, none have worked, and this is not covered in any of my lectures, which makes me assume this is considered a pre-requisite, which unfortunately I don't have in my background or don't recall.
I have spent over a day looking for a way on how to tackle this and the most relevant content I found is the Youtube video below, however, this was a F_x(x), rather than a joint pdf, and going from one to the other doesn't seem like something I should take for granted or assume is too easy - also, I can't be 100% certain this is the correct track, but I think so.
video I tried: https://www.youtube.com/watch?v=Vf7ER2GSenI
Could someone please advise how to approach this problem...
Thank you very much for the help! It is much appreciated.
Set
$$\begin{cases} u=xy \\ v=\frac{x}{y} \end{cases}\rightarrow\begin{cases} x=\sqrt{uv} \\ y=\sqrt{\frac{u}{v} } \end{cases}$$
it is self evident that it must be also
$$\begin{cases} 1\leq uv<+\infty ,& (1) \\ 1\leq \frac{u}{v} <+\infty,&(2) \end{cases}$$
Plotting this region in the 1st Quadrant of Cartesian Axes you get the requested solution
In fact (1)is satisfied under the condition $u\geq \frac{1}{v}$ while (2) is satisfied when $u\geq v$
When $v\in (0;1)$ the first condition (1) is stronger, when $v>1$ is the second one (2) to be the strongest