If $X$ and $Y$ are independent and identically distributed uniformly on $(0, 1)$, what is the joint density of the following?
$U = X$ and $V = X/Y$?
To compute this distribution, I use the Jacobian method. Since $X$ and $Y$ are independent, I know $f(x, y) |J|^{-1}= f_{X}(x) \cdot f_{Y}(y) |J|^{-1}$gives the probability distribution. So, I calculated it, and I got $U/V^{2}$, which is correct according to my solution's manual.
But what are the bounds on $U$ and $V$? How can I get the bounds? Like, for example for the $f_{X}$ and $f_{Y}$, I know $1$ if $0 \leq x \leq 1$ and $0 \leq y \leq 1$ and $0$ otherwise (because it's uniformly distributed). But what about after the transformation? How can I get that?
Best Answer
Solve for $X,Y$ in terms of $U,V$. You get $X=U$ and $Y=\frac U V$. The constraints are $0<U<1$ and $0<U<V$. So $u$ ranges from $0$ to $v$ and then $v$ from $0$ to $1$.