Integrate a function from 0 to 1 using the Sample Mean Monte Carlo algorithm
MC.samplemean(
FUN = function(x) x - x^2,
n = ani.options("nmax"),
col.rect = c("gray", "black"),
adj.x = TRUE,
...
)the function to be integrated
number of points to be sampled from the Uniform(0, 1) distribution
colors of rectangles (for the past rectangles and the current one)
should the locations of rectangles on the x-axis be adjusted?
If TRUE, the rectangles will be laid side by side and it is
informative for us to assess the total area of the rectangles, otherwise
the rectangles will be laid at their exact locations.
other arguments passed to rect
A list containing
the Uniform random numbers
function values evaluated at x
number of points drawn from the Uniform distribtion
the estimated value of the integral
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 \(\sum f(x_i) 1/n\), 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.
Examples at https://yihui.org/animation/example/mc-samplemean/