Transforming a pdf.

This is another unnamed result, but it seems pretty useful.

Let X be a random variable and Y = g(X) then

1. if g is a strictly increasing function on the support of X then F_Y(y) = F_X(g^-1(y)) for y in the support of Y. (The support of a continuous random variable is the set of values where the density function is greater than zero).
2. if g is a strictly decreasing function on the support of X then  F_Y(y) = 1 - F_X(g^-1(y)) 

If the g^-1(y) has a continuous derivative when g^-1(y) is in the support of X, then it immediately follows that:

f_Y(y) = f_X(g^-1(y)) |d/dy g^-1(y)| for all y in the support of Y (and zero elsewhere).

This can make quick work of useful, but otherwise non-trivial, transformations like the Inverted Gamma and Square of a Normal (which is Chi-squared with 1 degree of freedom).