Change of Variables 11/3

Theorem:
For $\phi$ 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:

$$\begin{eqnarray*}If \;\phi\; \mbox{ is increasing}& F_Y(y)&=F_X(\phi^{-1}(y)) \... ...& \mbox{ for some }x\\ 0, &\mbox{ otherwise} \end{array}\right. \end{eqnarray*}$$

I did several examples in lecture 11/4, the most important is the affine transformation:
Y=g(X)=aX+b, then $X=g^{-1}(Y)=\frac{Y-b}{a}$, whose derivative: $\frac{dg^{-1}(y)}{dy}=\frac{1}{a}$ gives the new density as:

$$f_Y(y)=\frac{1}{\vert a\vert}f_X(g^{-1}(y)=\frac{1}{\vert a\vert}f_X(\frac{y-b}{a})$$

This is the most important transformation.