Investigate the exponential distribution in R and compare it with the Central Limit Theorem

Overview

The purpose of this data analysis is to investigate the exponential distribution and compare it to the Central Limit Theorem. For this analysis, the lambda will be set to 0.2 for all of the simulations. This investigation will compare the distribution of averages of 40 exponentials over 1000 simulations.

Simulations

Set the simulation variables lambda, exponentials, and seed.

ECHO=TRUE
set.seed(1337)
lambda = 0.2
exponentials = 40

Run Simulations with variables

simMeans = NULL
for (i in 1 : 1000) simMeans = c(simMeans, mean(rexp(exponentials, lambda)))

Sample Mean versus Theoretical Mean

Sample Mean

Calculating the mean from the simulations with give the sample mean.

mean(simMeans)

## [1] 5.055995

Theoretical Mean

The theoretical mean of an exponential distribution is lambda^-1.

lambda^-1

## [1] 5

Comparison

There is only a slight difference between the simulations sample mean and the exponential distribution theoretical mean.

abs(mean(simMeans)-lambda^-1)

## [1] 0.05599526

Sample Variance versus Theoretical Variance

Sample Variance

Calculating the variance from the simulation means with give the sample variance.

var(simMeans)

## [1] 0.6543703

Theoretical Variance

The theoretical variance of an exponential distribution is (lambda * sqrt(n))^-2.

(lambda * sqrt(exponentials))^-2

## [1] 0.625

Comparison

There is only a slight difference between the simulations sample variance and the exponential distribution theoretical variance.

abs(var(simMeans)-(lambda * sqrt(exponentials))^-2)

## [1] 0.0293703

Distribution

This is a density histogram of the 1000 simulations. There is an overlay with a normal distribution that has a mean of lambda^-1 and standard deviation of (lambda*sqrt(n))^-1, the theoretical normal distribution for the simulations.

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.6.3

ggplot(data.frame(y=simMeans), aes(x=y)) + 
  geom_histogram(aes(y=..density..), binwidth=0.2, fill="steelblue",color="black") +
  stat_function(fun=dnorm, args = list(mean=lambda^-1, sd=(lambda*sqrt(exponentials))^-1), size=2, col = "palegreen") +
  labs(title="Plot of the Simulations", x="Simulation Mean")