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.