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Theorem:

For
a one-to-one function,
(you can recognize this by whether
it is always strictly
increasing or always strictly decreasing),
let *X*'s density be *f*_{X} and *Y*'s density
*f*_{Y} then:

I did several examples in lecture 11/4, the most important is
the affine transformation:

*Y*=*g*(*X*)=*aX*+*b*, then
,
whose derivative:
gives the new density as:

This is the most important transformation.

*Susan Holmes*

*1998-12-07*