The algorithm for this procedure is

• Use the uniform distribution to find the random variable $y$ in the interval [0,1]. The function $p(x)$ is a user provided PDF.
• Evaluate thereafter

$$\begin{equation*} I=\int_a^b F(x) dx =\int_a^b p(x)\frac{F(x)}{p(x)} dx, \end{equation*}$$ by rewriting $$\begin{equation*} \int_a^b p(x)\frac{F(x)}{p(x)} dx = \int_{\tilde{a}}^{\tilde{b}}\frac{F(x(y))}{p(x(y))} dy, \end{equation*}$$ since $$\begin{equation*} \frac{dy}{dx}=p(x). \end{equation*}$$

• Perform then a Monte Carlo sampling for

$$\begin{equation*} \int_{\tilde{a}}^{\tilde{b}}\frac{F(x(y))}{p(x(y))} dy\approx \frac{1}{N}\sum_{i=1}^N\frac{F(x(y_i))}{p(x(y_i))}, \end{equation*}$$ with $y_i\in [0,1]$,

• and evaluate the variance as well.