Improved Monte Carlo Integration

We have presented a simple brute force approach to integration with the Monte Carlo method. There we sampled over a given number of points distributed uniformly in the interval $[0,1]$ $$\begin{equation*} I=\int_0^1 f(x)dx=\langle f\rangle. \end{equation*}$$ Here we introduce two important steps which in most cases improve upon the above simple brute force approach with the uniform distribution, namely

• change of variables and
• importance sampling