# Calc-Based Statistics: Transformation of Two R.V.s, Change-of-Variables

1. Apr 9, 2012

### aleph_0

So I have thoughts about how to solve this problem, but more than anything I want a bit of a sanity check on what I've done so far, so that I don't spend the rest of tonight down a wrong path.

The problem:

Let the joint pdf of $X,Y$ be

$\frac{1}{x^{2}y^{2}}, \quad x \geq 1, \quad y \geq 1$

Compute the joint pdf of $U = XY, \quad V = X/Y$.

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My solution so far: $uv = x^{2}, \quad u/v = y^{2}$ and since $x,y \geq 1$ then $x = \sqrt{uv}, \quad y = \sqrt{u/v}$. Then compute the Jacobian.

From this I trace out the boundaries of these two transformations by first setting $x$ to 1 and seeing how $u$ varies as $y$ ranges from 1 to infinity, etc.

Is this all sound and the smartest way to solve the problem?