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

[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 [itex]X,Y[/itex] be

[itex]\frac{1}{x^{2}y^{2}}, \quad x \geq 1, \quad y \geq 1[/itex]

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

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

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

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

 

[Jhero]

Yes, that's a good approach to the problem. You can also use the change of variables formula to compute the joint pdf of U and V. The formula is f_{U,V}(u,v) = f_{X,Y}(x(u,v), y(u,v))|J|,where J is the Jacobian of the transformation (which you have already computed).