---
title: MC.samplemean()
subtitle: Sample Mean Monte Carlo Integration
date: '2017-04-04'
slug: 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.
 
```{r demo-a, cache=TRUE, interval=.1}
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

integrate(function(x) x - x^2, 0, 1)

```


```{r demo-b, cache=TRUE, interval=.1}

## 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

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