This is a comparison of a distribution of the averages of exponential samples. The mean and variance will be compared to theoretical values and the distribution will be compared to a normal distribution.

Simulations

Before comparisons to mean, variance and overall distribution can be made, simulations must be ran. Lambda will be set to 0.2, the average of 40 exponentials will be taken and this simulation will run for 1000 iterations.

Sample Mean versus Theoretical Mean

Mean Simulation

MExp = NULL
set.seed(33)
for (i in 1:1000) MExp = c(MExp, mean(rexp(40,0.2)))

Mean Histogram

Sample Variance versus Theoretical Variance

Variance Simulation

MVar = NULL
set.seed(33)
for (i in 1:1000) MVar = c(MVar, (MExp[i]-mean(MExp))*sqrt(40)/(1/.2))

Variance Histogram

The sample is very close to the theoretical value.

Distribution

Simulation of 1000 AVERAGE Exponential Distributions

AExp = NULL
set.seed(33)
for (i in 1:1000)
  AExp = c(AExp, mean(rexp(40,.2)))

The collection of averages much more closely resembles a normal distribution than a random collection of 1000 exponential samples.