With this definition of a random variable and its associated PDF, we attempt now a clarification of the Monte-Carlo strategy by using the evaluation of an integral as our example.

In discussion on numerical integration we went through standard methods for evaluating an integral like $$\begin{equation*} I=\int_0^1 f(x)dx\approx \sum_{i=1}^N\omega_if(x_i), \end{equation*}$$ where $\omega_i$ are the weights determined by the specific integration method (like Simpson's method) with $x_i$ the given mesh points. To give you a feeling of how we are to evaluate the above integral using Monte-Carlo, we employ here the crudest possible approach. Later on we will present slightly more refined approaches. This crude approach consists in setting all weights equal 1, $\omega_i=1$. That corresponds to the rectangle method $$\begin{equation*} I=\int_a^bf(x) dx \approx h\sum_{i=1}^N f(x_{i-1/2}), \end{equation*}$$ where $f(x_{i-1/2})$ is the midpoint value of $f$ for a given value $x_{i-1/2}$.