Transformations of probability distributions?

[hushed cloak]

Essentially this is a problem I stumbled on in a disparate field.

The basic statement of the problem is this:

There are two probability distributions f and g with the domain being [0, 1]. By definition they are normalized.

What is a transformation M, or a class of such transformations that maps f to g.

It would also be helpful to get pointers on where to look, because I started Theory of Distributions by Svetlin G. Georgieve and it is quite intimidating. I can put my head down and get cracking, but I'd like to ensure there's no better alternatives before that.

Any help would be appreciated!

[turbid sky]

We can use inverse transform sampling for this.
Suppose you have a random variable X with CDF F(x). Then F(X) ~ U[0, 1], regardless of the distribution of X.
So, suppose you have two random variables: X1 and X2. Let their CDFs be F(X1, x) and F(X2, x) and their quantile functions be Q(X1, p) and Q(X2, p), respectively. Then by the above we have:
F(X1, X1) ~ U[0, 1]
F(X2, X2) ~ U[0, 1]
So, F(X1, X1) and F(X2, X2) have the same distribution. Thus, if we take the quantile function of one of the variables of both sides, we get the following statements:
X1 has the same distribution as Q(X1, F(X2, X2))
X2 has the same distribution as Q(X2, F(X1, X1))
Thus, to get from X1 to X2 you apply the fuction φ(x) = Q(X2, F(X1, x)), and to get from X2 to X1 you apply its inverse ψ(x) = Q(X1, F(X2, x)).

[hushed cloak]

What is the justification for F(X) ~ U[0,1]? Assuming U[0,1] is a uniform distribution, correct?

[turbid sky]

Let's see. Suppose Y = F(X, X). Then:
F(Y, x) = P(Y < x) = P(F(X, X) < x) = P(X < Q(X, x)) = F(X, Q(X, x))
As the domain of Q(X, x) is (0, 1) and F(X, x) and Q(X, x) are inverses, we get:
F(Y, x) = x, 0 < x < 1
As F(Y, x) is a CDF, we must have F(Y, -∞) = 0 and F(Y, +∞) = 1. But we already have F(Y, 0+) = 0 and F(Y, 1-) = 1, so we get:
F(Y, x) = ...
...0, x ≤ 0
...x, 0 < x < 1
1, x ≥ 1
But this is precisely the CDF of U[0, 1]. So, regardless of the initial distribution, we proved that F(X, X) ~ U[0, 1].

As you can imagine, this is extremely useful, as this allows you to generate values from any distribution if you can generate uniform ones, which is somwthing a computer can do.

[hushed cloak]

Thanks! I'll take some time to get my teeth sunk into this. Might pop back in if I have questions.