Probability distribution for a function of a random variable

Question

I have the distribution of X with respect to parameter t vaying between 0 and 1. However, in nature, parameter t is not uniformly distributed. It has a known probability distribution. What is distribution of X given distribution of t? The product? Can you clarify, and hint at notions or concepts involved?

EDIT: in fact, problem can be formulated this way: X is a function of t,

$X: t -> X(t)$

and I would like to have an expression for P(X(t)) when I known the distribution for t.

Thanks

Answer 1

Let t have density f and let x=g(t). If g is 1-1 the t=g$^-$$^1$(x). Substitute g$^-$$^1$(x) for t in f and then multiply by the Jacobian of the transformation to get the density for x.