# computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ where for simplicity I choose $X,Y \sim B(2,2)$, independent. This represents the beta random variable with both parameters equal to 2.

Now, we know that Area of $\Gamma$ $= \displaystyle \iint_{\Gamma}\,dx\,dy=\iint_{\Omega} 1_{(x,y) \in \Gamma} \,dx\,dy = \iint_{\Omega} 1_{(x,y) \in \Gamma} \frac{1}{f(x)f(y)} f(x)f(y) \,dx \,dy = E \left[1_{(x,y) \in \Gamma} \frac{1}{f(x)f(y)} \right]$

where $\Omega = [0,1]^2$ and $f$ is the marginal pdf of $B(2,2)$.

By Monte Carlo, computing $E \left[1_{(x,y) \in \Gamma} \frac{1}{f(x)f(y)} \right]$ ought to be straightforward.

But I have a few concerns:

1) $f(x) = 0$ if $x = 0,1$. It is possible that $\Gamma$ is say, the square $[0,0.5]^2$ which means $\frac{1}{f(x)f(y)}$ will be undefined theoretically. BUT we know that from a simulation point of view, since I will never be able to obtain a beta distributed RV realization of 0 in MATLAB, I should have no computational problems. Am I violating anything with this approach?

2) Secondly, is there an alternative way to do this? Any links to papers, etc., which have dealt with this before?

I'd appreciate any insight. Thanks!

• 1) is fine. If you are worried about throwing an exception, add machine epsilon to f; it won't affect the precision of the final result. 2) too broad. Of course there are other ways to estimate areas. It depends on what you know about the region. – user147263 Jun 11 '15 at 22:16