The Change-of-Variables Method for Continuous Random Variables

The distribution function method finds the density of $Y = g(X)$ by first computing $F_Y(y) = P(Y \le y)$ and then differentiating. When $g$ is strictly monotonic and differentiable on the support of $X,$ we can skip the CDF entirely and write the density of $Y$ directly.

Change-of-variables formula. Let $X$ be continuous with density $f_X,$ and let $g$ be strictly monotonic and differentiable on the support of $X,$ with inverse $g^{-1}.$ Then $Y = g(X)$ has density

$$f_Y(y) = f_X\!\left(g^{-1}(y)\right)\left|\dfrac{d}{dy}\,g^{-1}(y)\right|$$

on the image of the support of $X$ under $g,$ and $f_Y(y) = 0$ elsewhere.

For example, let $X$ have density $f_X(x) = 4x^3$ on $0 < x < 1,$ and let $Y = 5X.$ Then $g(x) = 5x,$ so $g^{-1}(y) = \dfrac{y}{5}$ and $\dfrac{d}{dy}\,g^{-1}(y) = \dfrac{1}{5}.$ The support transforms from $(0,1)$ to $(0,5),$ giving

$$f_Y(y) = 4\left(\dfrac{y}{5}\right)^{\!3}\cdot\dfrac{1}{5} = \dfrac{4y^3}{625}, \quad 0 < y < 5.$$