1 Overview
We show that the distribution of averages of exponentially distributed3 variables becomes that of a standard normal as the sample size increases (according to the Central Limit Theorem (CLT)4).
2 Simulations
Let us first fix some parameters, including the sample size, the number of simulations, the rate of our exponential distribution.
n <- 40 # sample size
lambda <- 0.2 # the rate of the exponential distribution
t <- 1000 # number of simulations
mu <- 1/lambda # the mean of the distribution
sigma <- 1/lambda # the standard deviation of the distributionThe following script simulates 1000 (t) sample means with size 40 (n). The result is stored into a variable named mns.
mns <- NULL
for(i in 1:t) mns = c(mns, mean(rexp(n, lambda)))3 Sample Mean versus Theoretical Mean
The sample mean, i.e., mean(mns) \(\approx\) 4.96, and the theoretical mean, i.e., mu = 5, are almost equal. The following plot shows that as number of simulations increases, the sample mean (in \(\color{blue}blue\)) converges to the theoretical mean (in \(\color{red}red\)).
cummns <- cumsum(mns)/(1:t)
plot(cummns, type="l", lwd=2, col = "blue",
cex=1, cex.lab = .7, cex.axis = .6, cex.main = .7, cex.sub=.5,
main = "Sample Mean versus Theoretical Mean",
xlab = "Number of simulations", ylab = "Cumulative Mean")
abline(h=sigma, col="red", lwd=1)
4 Sample Variance versus Theoretical Variance
The sample variance, i.e, var(mns) \(\approx\) 0.6, and the theoretical variance, sigma^2/n = 0.625, are almost the same. The following plot shows that as number of simulations increases, the sample variance converges to the theoretical variance:
vars <- cumsum(mns^2)/(1:t)-cummns^2
plot(vars, type="l", lwd=2, col = "blue",
cex=1, cex.lab = .7, cex.axis = .6, cex.main = .7, cex.sub=.5,
main = "Sample Variance versus Theoretical Variance",
xlab = "Number of Simulations", ylab = "Cumulative Variance")
abline(h=sigma^2/n, col="red", lwd=1)
5 Distribution
Let us denote the distribution of sample means by \(\bar{X}_n\), where \(n\) denotes the sample size. According to the CLT, \(\bar{X}_n\) ~ \(N(\mu, \sigma^2/n)\), where \(\mu\) and \(\sigma\) are the mean and standard deviation of the distribution, respectively. We are going to show that this holds in our case, where \(\mu = 5\), \(\sigma = 5\), and \(n = 40\).
The following script visualizes the density of our sample mean (denoted by a \(\color{blue}blue\) line), and the normal distibution \(N(\mu, \sigma^2/n) = N(5, 0.625)\) (denoted by a \(\color{red}red\) line). The densities are shown over a histogram of our simulation, mns. The blue and red straight lines indicate where the mean of the distributions are, respectively (they almost overlap). As we see in the figure, our simulation makes a good approxmiation of the corresponding normal distribution.
hist(mns, density = 20, breaks = 20, prob = TRUE,
cex.lab = .8, cex.axis = .8, cex = 1, cex.main = .7,
main = "The Distribution of Averages of 40 Exponentials", xlab = "means")
lines(density(mns), lwd = 2, col = "blue")
curve(dnorm(x, mean = mu, sd = sigma/sqrt(n)), col = "red", lwd = 3, add = TRUE)
abline(v = mean(mns), col = "blue")
abline(v = mu, col = "red")
Email: a.a.safilian@gmail.com↩
This analysis report is the final project-Part1 of the Statistical Inference (Coursera) course at Johns Hopkins University↩