next up previous index
Next: Exponential Random Variable 11/4 Up: Special Continuous Distributions 11/2 Previous: Special Continuous Distributions 11/2

Change of Variables 11/3

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 fX and Y's density fY then:

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

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:

\begin{displaymath}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.

Susan Holmes