Overview

In this project you will investigate the exponential distribution in R and compare it with the Central Limit Theorem. 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. You will investigate the distribution of averages of 40 exponentials. Note that you will need to do a thousand simulations.

Simulations

Simuation of exponential distribution with following data in R with rexp(n, lambda),lambda = 0.2 and exponentials (n = 40) for thousand simulations.

Set Seed , size , lambda

set.seed(100)
n <- 40
lambda <- 0.2

Set No. of simulation according to samples of size 40 in thousand columns and 40 rows

simulations <-  1000
sim_exp <- replicate(simulations, rexp(n, lambda))

Calculate mean of each column

mean_sim_exp <- apply(sim_exp, MARGIN = 2, mean)
mean_analytical<-- mean(mean_sim_exp)
mean_analytical

Evaluation of first 10 mean of thousands simulation

head(mean_sim_exp, 10)

[1] 4.137412 6.051703 4.415869 4.404714 3.210413 5.475307 4.405938 6.573635 5.399291 [10] 4.913581

Calculation of Sample & Theoretical Means

Sample_mean<-mean(mean_sim_exp)
Sample_mean

[1] 4.999702

mean_theory <- 1/lambda
mean_theory

[1] 5 #Histogram

hist(mean_sim_exp, xlab = "mean", main = "Exponential Simulations",col="green")

Showing sample mean

abline(v = Sample_mean, col = "red")
abline(v = mean_theory, col = "blue")

Calculation of sample & theory variance

 sd_sim_exp<-sd(sim_exp)
  sterror_sim_exp<-sd(sim_exp)/sqrt(n)
 sterror_sim_exp

[1] 0.7965831

Distribution 40 Simulated exponentials mean Vs normal distribution mean

qqnorm(mean_sim_exp)
qqline(mean_sim_exp, col = 2)

Conclusion

Distribultion of the central limit theorem is approximately close to 40 simulated exponentials