Note the change in integration limits from $a$ and $b$ to $\tilde{a}$ and $\tilde{b}$. The advantage of such a change of variables in case $p(y)$ follows closely $F$ is that the integrand becomes smooth and we can sample over relevant values for the integrand. It is however not trivial to find such a function $p$. The conditions on $p$ which allow us to perform these transformations are

• $p$ is normalizable and positive definite,
• it is analytically integrable and
• the integral is invertible, allowing us thereby to express a new variable in terms of the old one.