With the aid of the above variable transformations we address now one of the most widely used approaches to Monte Carlo integration, namely importance sampling.

Let us assume that \( p(y) \) is a PDF whose behavior resembles that of a function \( F \) defined in a certain interval \( [a,b] \). The normalization condition is $$ \begin{equation*} \int_a^bp(y)dy=1. \end{equation*} $$ We can rewrite our integral as $$ \begin{equation*} I=\int_a^b F(y) dy =\int_a^b p(y)\frac{F(y)}{p(y)} dy. \tag{20} \end{equation*} $$