Exponential Distribution Simulation

Overview

This project is for the course Statistical Inference on Coursera. It simulates the exponential distribution in R. The simulation compare the sample mean and theoretical mean, sample variance and theoretical variance, and show the distribution of sample mean is approximately normal.

Simulation

Ramdonly initial sample data.

set.seed(1)
n <- 1000; sample_size <- 40
lambda <- 0.2
mns = NULL
vars = NULL
for (i in 1:n){
    exp_temp <- rexp(sample_size, rate=lambda)
    mns = c(mns, mean(exp_temp))
    vars = c(vars, var(exp_temp))}

Sample Mean versus Theoretical Mean

library(ggplot2)
means <- cumsum(mns) / (1:n)
g <- ggplot(data.frame(x = 1:n, y=means), aes(x=x, y=y) )
g <- g + geom_hline(yintercept = 5) + geom_line(size = 2)
g <- g + labs(x = "Number of obs", y = "Cumulative mean")
print(g); means[n]

## [1] 4.990025

Sample mean is 4.990025, theoretical mean is 5.

Sample Variance versus Theoretical Mean

variances <- cumsum(vars) / (1:n)
p <- ggplot(data.frame(x = 1:n, y=variances), aes(x=x, y=y) )
p <- p + geom_hline(yintercept = 25) + geom_line(size=2)
p <- p + labs(x = "Number of obs", y= "Cumulative variance")
print(p); variances[n]

## [1] 25.06459

Sample variance is 25.06459, theoretical variance is 25.

Show the distribution is approximately normal

x <- seq(min(mns), max(mns), length=1000)
y <- dnorm(x, mean = 1/lambda, sd = 1/lambda/sqrt(sample_size))
plotdata <- data.frame(mns, x, y)
q <- ggplot(data.frame(plotdata), aes(x=mns) )
q <- q + geom_histogram( aes(y=..density..), fill = "lightblue", binwidth=0.2, colour = "black")
q <- q + geom_density(colour="black", size=2)
q <- q + geom_line(aes(x=x, y=y), colour="red", size=1, linetype=2)
q <- q + geom_vline(xintercept=5, colour="red", size=1, linetype=2)
print(q); qqnorm(mns); qqline(mns)