Generating a uniform distribution (inverse-transformation)

Question

Assume that I can generate samples from a continuous random variable Y with an invertible cdf FY. I wonder what is the distribution of FY (Y ) (capital Y both inside and outside)?

My primary guess is a uniform distribution, but I'm not sure how to validate my answer.

Answer 1

This is a common homework problem so I think an outline of the approach is called for.

Let F be the cdf of Y.

Let U = F(Y).

Now calculate the cdf of U: P(U <= u)

Substitute in a function of Y, get that in terms of P(Y <= something) and recognize that P(Y <= something) can be written as F(something) then simplify and recognize the cdf of U (or take the cdf back to a density).

The detailed derivation is given at Wikipedia but it's much better to do it yourself by hand as above -- you'll actually have a chance to remember it when you need it.