The Distribution Function Method

The distribution function method finds the distribution of $Y = g(X)$ by working directly from the definition of the CDF:

$$F_Y(y) = P(Y \le y) = P(g(X) \le y).$$

The procedure has three steps:

1. Rewrite the event $\{g(X) \le y\}$ as an equivalent event involving $X$ alone.
2. Evaluate the resulting probability using $F_X$.
3. Differentiate to recover the pdf: $f_Y(y) = F_Y'(y)$.

To illustrate, let $X \sim U(0, 1)$ so $F_X(x) = x$ on $[0,1]$, and let $Y = X^3$. Since $g(x) = x^3$ is increasing on this interval,

$$F_Y(y) = P(X^3 \le y) = P(X \le y^{1/3}) = F_X(y^{1/3}) = y^{1/3}, \quad 0 \le y \le 1.$$

Differentiating,

$$f_Y(y) = \dfrac{1}{3}\, y^{-2/3}, \quad 0 < y \le 1.$$