MC.samplemean()

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

plot of chunk demo-a

## [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

plot of chunk demo-b

## [1] 0.09573

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

## 0.125 with absolute error < 2.4e-15