Sample Mean Monte Carlo Integration

As discussed before, in physics and statistics many of the problems Monte Carlo is used on is under the form of the estimate of an integral unkown in closed form: $\hat{\theta}=\int_0^1 f(u)du$, which can be seen as the evalutaion of $Ef(u)$, where $u \sim U(0,1)$. There is another Monte Carlo method named mean-value Monte Carlo method. Thus proposes to generate $B$ numbers uniformly from $(0,1)$ and take their average: to estimate $\theta$, $B=\frac{1}{B}\sum\limits_{b=1}^{B}f(u_b)$.

In the animation package, the function MC.samplemean() can be used to estimate the area under the curve, by computing the average function values. Let’s me show you a quick example, see below.

library(animation)
MC.samplemean(border = NA)$est

## [1] 0.1815

Reference

1.Monte Carlo,http://www-stat.stanford.edu/~susan/courses/s208/node14.html