I'm taking a course in probability and struggling with transformations of random variables. An example in the text is:
Let U1 and U2 be independent uniform random variables, both on (0,1). Let Y1 = U1U2 and let Y2=U2.
(a) Find the joint density of Y1 and Y2.
If I calculate the inverse functions for U1 and U2, I get:
U1 = Y1/Y2
U2 = Y2
But I feel like I'm already on shaky ground because Y1 and Y2 defined over (0,1), which makes U1 undefined at Y2=0. Regardless, calculating the Jacobian gives my 1/Y2 - again, feels like shaky ground.
From this, the joint density should be 1/Y2 since this is defined as the product of the densities of U1 and U2 (uniform over (0,1), so densities equal 1) and the Jacobian is 1/Y2.
(b) What is the marginal density for Y1?
If I take the integral from 0 to 1 of the density above, I get -infinity. The integral of 1/Y2 is log(Y2), and I have to find that at 0 for the lower limit of the integral.
This seems like it should be an easy problem, but I can't wrap my head around where I've gone wrong.