Editing Monte Carlo integration (section)

=== Example ===
[[File:Relative error of a Monte Carlo integration to calculate pi.svg|thumb|right|350px|Relative error as a function of the number of samples, showing the scaling <math>\tfrac{1}{\sqrt{N}}</math>]]

A paradigmatic example of a Monte Carlo integration is the estimation of π. Consider the function
<math display="block">H\left(x,y\right)=\begin{cases}
1 & \text{if }x^{2}+y^{2}\leq1\\
0 & \text{else}
\end{cases}</math>
and the set Ω = [−1,1] × [−1,1] with ''V'' = 4. Notice that
<math display="block">I_\pi = \int_\Omega H(x,y) dx dy = \pi.</math>

Thus, a crude way of calculating the value of π with Monte Carlo integration is to pick ''N'' random numbers on Ω and compute
<math display="block">Q_N = 4 \frac{1}{N}\sum_{i=1}^N H(x_{i},y_{i})</math>

In the figure on the right, the relative error <math>\tfrac{Q_N-\pi}{\pi}</math>  is measured as a function of ''N'', confirming the <math>\tfrac{1}{\sqrt{N}}</math>.