Method of Distribution Functions

1. Find $y$ bounds: e.g. $F_U(u)=P(U\le u)=P(3y+1\le u)=P(y\le\frac{u+1}{3})$
2. Find $u$ range: e.g. $0 \le \frac{u+1}{3} \le 1$
3. Integrate for CDF: e.g. $F_U(u)=\int_{-\infty}^{\frac{u+1}{3}} 2y \, dy$
4. Differentiate for PDF: e.g. $\frac{d}{du} (\frac{u+1}{3})^2$
5. Write piecewise solution

Transformation Method

Let $Y$ have a PDF (density) $f_Y(y)$. If $h(y)$ is incr/decr for all $y$ with $f_Y(y)>0$

then, for $U=h(Y)$:

$f_U(u)=f_Y(h^{-1}(u))|\frac{dh^{-1}}{du}|$