Overview

In this document, we are going to show the comparison between exponential distribution and its central limit. The exponential distribution can be simulated in R with rexp(n, 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. We will investigate the distribution of averages of 40 exponentials and do a thousand simulations.

Simulation

Setting the seed for the pseudo-random number generator and also ensure reproducible simulation.

library(ggplot2)
set.seed(19830)
data = replicate(1000,mean(rexp(40,0.2)))
df = data.frame(data)
colnames(df) <- c("mean")
p <- ggplot(df,aes(x=mean))+labs(x="",y="")+geom_histogram(colour="black", fill="white")+ggtitle("Exponential Distribution") 
p

Sample Mean versus Theoretical Mean

The theoretical mean of exponential distribution is

1/0.2
## [1] 5

The sample mean of the simulated data is

mean(data)
## [1] 4.992918
p + geom_vline(aes(xintercept=mean(data)), color="red", linetype="dashed", size=1)+
  geom_vline(aes(xintercept=1/0.2), color="blue", linetype="dashed", size=1)

Sample Variance versus Theoretical Variance

The theoretical variance is

1/(0.2*sqrt(40))
## [1] 0.7905694

The sample variance of the simulated data is

var(data)
## [1] 0.6495278

Distribution:

We can apply qqplot to show how close the simulated distribution and normal distribution. Normal distribution appears to be good approximation to simulated exponential distribution due to central limit theorem.

qqnorm(data)
qqline(data)