simulation of distribution of means draw from exponential distribution

"central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution." [1]

I am going to draw 40 numbers from exponential distribution [2] (you can do it for any distribution) for 1000 times and examine the distribution of the means. In R, you can do it by rexp(40, lambda),  where lambda is the rate parameter. The mean of exponential distribution is 1/lambda and the standard deviation is also 1/lambda. Set lambda = 0.2 for all of the simulations for this specific test.

see a gist below.

	## Overview
	# central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.
	# I am going to draw 40 numbers from exponential distribution for 1000 times, and calcuate the mean
	# of each draw (we will have 1000 means), and through this simulation to test if the
	# distribution of the means will be normal or not.
	## start simulation
	# number of simulation, sample size and lambda
	nosim<- 1000
	n<- 40
	lambda<- 0.2
	# make a matrix containing 1000 rows and 40 columns with simulated value
	mat<- matrix(rexp(nosim * n, lambda), nosim)
	# calculate the mean of the 40 numbers draw from exponential distritubtion
	# for each row (total 1000 rows), and then take the mean
	# we got 1000 means of the 1000 simulation
	# now let's calculate the standard deviation
	mean_expo<- apply(mat, 1, mean)
	## the sample mean and compare it to the theoretical mean of the distribution
	#sample mean
	mean(mean_expo)
	# 5.1
	# theroretical mean
	1/lambda
	# 5.017575
	# they are very close to each other.
	## how variable the sample is (via variance) and compare it to the theoretical variance of the distribution?
	# sample variance
	var(mean_expo)
	# 0.6469216
	# thoretical variance
	(1/lambda)^2 * 1/n
	# 0.625
	# they are very close to each other. you may get a different mean and variance for the sample, but if I set the seed. it should be reproducible
	# Let's look at the distribution of the means by histogram
	library(ggplot2)
	g<- ggplot(data.frame(mean_expo)) + geom_histogram(aes(x=mean_expo, y=..density..),
	fill="black", color="white")
	g<- g + geom_vline(xintercept=5, color="red")
	g<- g + geom_vline(xintercept= mean(mean_expo), color="green")
	g<- g + stat_function(fun = dnorm, args= list(mean= 1/lambda, sd= 1/lambda * 1/sqrt(40)),
	color="red")
	g<- g + geom_density(aes(x= mean_expo,y=..density..), color="green")
	g
	# the distribution of means should be centered on 1/lambda = 1/0.2 = 5 (the theoretical center).
	# the theoretical standard deviation should be sigma/sqrt(n).
	# the red bell curve is the density function of Normal(1/lambda, 1/lambda * 1/sqrt(40)).
	# the red vertical line is the theoretical mean (5).
	# the green vertical line is the sample mean(5.036)
	# the green bell curve is the density of the simulated sample means.
	# we can see they are very close to each other.
	## compare with the large number of exponential distribution
	expo<- rexp(1000, lambda)
	g<- ggplot(data.frame(expo)) + geom_histogram(aes(x=expo, y=..density..),
	fill="black", color="white")
	g<- g + geom_vline(xintercept=5, color="red")
	g<- g + stat_function(fun = dnorm, args= list(mean= 1/lambda, sd= 1/lambda * 1/sqrt(40)),
	color="red")
	g<- g + geom_density(aes(x= expo,y=..density..), color="green")
	g
	# clearly, we see it is not normal distributed

view raw simulation.r hosted with ❤ by GitHub

[1] http://en.wikipedia.org/wiki/Central_limit_theorem

[2] http://en.wikipedia.org/wiki/Exponential_distribution