Simulation Experiment of exponential distributions

Overview

Simulation of the exponential distribution to find results about its mean and standard deviation.

For mean

library(ggplot2)
set.seed(2000)
n = 40
lambda = .2
nosim = 1000
binwidth <- 0.5
mean_list <- numeric()
## Finding sample mean
for(i in 1:nosim){
        exp <- rexp(n, lambda)
        mean_list <- c(mean_list, mean(exp))
}
theoretical_mean_averages = 1/lambda
theoretical_sd_averages = 1/(lambda*sqrt(n))

g <- qplot(mean_list) + geom_histogram(bins=10,binwidth = binwidth) + geom_vline(xintercept = theoretical_mean_averages, show.legend = T)
g <- g + xlab("Mean of averages of 40 random exponentials") + ylab("Frequency")
g <- g + stat_function( 
    fun = function(x, mean, sd, n, bw){ 
      dnorm(x = x, mean = mean, sd = sd) * n * bw
    }, 
    args = c(mean = theoretical_mean_averages, sd = theoretical_sd_averages, n = nosim, bw = binwidth))
g

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

We can see that the mean is approximately normally distributed about its theoretical mean which is 1/lambda = 1/0.2 = 5

The standard deviation of the exponential distribution is 1/lambda = 5 The standard deviation of the distribution of the mean of 40 random exponentials should be the standard error and that is the S/sqrt(40) = 5/sqrt(40)

Theoretical variance of the distribution of averages

theoretical_variance_averages = (1/(lambda*sqrt(40)))^2
theoretical_variance_averages

## [1] 0.625

Calculated variance of the simulations

var(mean_list)

## [1] 0.6168082

Therefore the two are approximately equal

Plot of a distribution of large number of exponentials vs the plot of the distribution of averages of 40 random exponentials

library(ggplot2)
library(cowplot)

## 
## Attaching package: 'cowplot'

## The following object is masked from 'package:ggplot2':
## 
##     ggsave

exp2 <- rexp(nosim, lambda)
q <- qplot(exp2) + geom_histogram(binwidth = 0.5, bins=30)
q <- q + xlab("Random exponentials") + ylab("Frequency")
plot_grid(q, g)

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.