Statistical Inference Course Project 1

Comparison of Exponential Distribution and Central Limit Theorem

Introduction

library(ggplot2)

## Warning: package 'ggplot2' was built under R version 3.2.3

Simulation

set.seed(0901)
n <- 40
lmd <- 0.2
rep <- 1000

simulations <- replicate(rep, rexp(n, lmd))

Sample Mean versus Theoretical Mean

each.mean <- apply(simulations, 2, mean)
real.mean <- mean(each.mean)
real.mean

## [1] 4.981791

theo.mean <- 1/lmd
theo.mean

## [1] 5

hist(each.mean, xlab = "Mean of Simulations", main = "Simulated Exponential Functions")
abline(v = real.mean)
abline(v = theo.mean)

Sample Variance versus Theoretical Variance

real.sd <- sd(each.mean)
real.var <- real.sd^2
real.sd

## [1] 0.7564694

real.var

## [1] 0.572246

theo.sd <- 1/lmd/sqrt(n)
theo.var <- theo.sd^2
theo.sd

## [1] 0.7905694

theo.var

## [1] 0.625

Distribution

means <- seq(min(each.mean), max(each.mean), length = rep)
density <- dnorm(means, mean = theo.mean, sd = theo.sd)

hist(each.mean, breaks = n, prob = TRUE, main = "Density of Means")
lines(means, density)

qqnorm(each.mean)
qqline(each.mean)