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.