Sample Mean Monte Carlo Integration

Yihui Xie & Lijia Yu / 2017-04-04

Integrate a function from 0 to 1 using the Sample Mean Monte Carlo algorithm

Sample Mean Monte Carlo integration can compute

$$I=\int_0^1 f(x) dx$$

by drawing random numbers $$x_i$$ from Uniform(0, 1) distribution and average the values of $$f(x_i)$$. As $$n$$ goes to infinity, the sample mean will approach to the expectation of $$f(X)$$ by Law of Large Numbers.

The height of the $$i$$-th rectangle in the animation is $$f(x_i)$$ and the width is $$1/n$$, so the total area of all the rectangles is $$\sum_{i=1}^{n}\frac{1}{n}f(x_i)$$, which is just the sample mean. We can compare the area of rectangles to the curve to see how close is the area to the real integral.

library(animation)
ani.options(interval = 0.2, nmax = 50)
par(mar = c(4, 4, 1, 1))

## when the number of rectangles is large, use border = NA
MC.samplemean(border = NA)$est  ## [1] 0.1638  integrate(function(x) x - x^2, 0, 1)  ## 0.1667 with absolute error < 1.9e-15  ## when adj.x = FALSE, use semi-transparent colors MC.samplemean(adj.x = FALSE, col.rect = c(rgb(0, 0, 0, 0.3), rgb(1, 0, 0)), border = NA)  ## another function to be integrated MC.samplemean(FUN = function(x) x^3 - 0.5^3, border = NA)$est

## [1] 0.09573

integrate(function(x) x^3 - 0.5^3, 0, 1)

## 0.125 with absolute error < 2.4e-15