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
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