Overview

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.

In this project, we are tasked two parts.

Exercise

Loading add-on package

library(ggplot2)

Set variables and seed

set.seed(1)
lambda <- 0.2
n <- 40
num_sim <- 1000

Make samples

#exp_data <- rexp(num_sim, lambda)    #the exponential distribution
mean_data <- NULL
for (i in 1 : num_sim) mean_data= c(mean_data, mean(rexp(n, lambda)))    #The mean of exponential distribution

1. Show the sample mean and compare it to the theoretical mean of the distribution.

The mean of the exponential distribution is 4.990025.

mean(mean_data)
## [1] 4.990025

The theoretical mean is 5.

1 / lambda
## [1] 5

2. Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution.

The sample variance of mean X is 0.6111165.

var(mean_data)
## [1] 0.6111165

The theoretical variance is 0.625.

(1 / lambda ^ 2) / n
## [1] 0.625

3. Show that the distribution is approximately normal.

g <- ggplot(data = data.frame(mean_data), aes(x=mean_data))
g + geom_histogram(aes(y = ..density..), bins = 40) +
  stat_function(fun = dnorm, args = list(mean = 1 / lambda, sd = sqrt(1 / lambda ^2 / n))) +
  xlab(expression(bar(X))) +
  ggtitle("Comparison of sample distribution and theoretical")

The above figure shows the distribution of sample mean is matched with a normal distribution.