Here I am going to 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. Here we take lambda equal to “0.2”.Now I go question wise.

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

We are taking 1000 samples and 40 as the size of the one sample.So that we will have 1000 mean one from the each sample.

#As we know that theoretical mean of exponential distribution is 1/lambada ie.1/0.2= 5.


theomean<-1/0.2
theomean
## [1] 5
sampsize<-40
numsamp<-1000
lambada<-0.2

First we see hoW “Exponential Dstribution”" looks like.I am taking here of size 40.In The Figure we can see the exponential nature of the histogram.

set.seed(11)
exp1<-rexp(1000,0.2)
hist(exp1,breaks =50,col="green",probability = T)
lines(density(exp1),lwd=3,col="blue")

Now we Take 1000 samples of size 40 and plot the distribution of means of each samples

set.seed(12)
means<-NULL
for(i in 1:numsamp){
  means<-c(means,mean(rexp(sampsize,lambada)))
}

hist(means,breaks=50,col="blue",main = "Distribution of 1000 sample means")

Now we will compare the theoretical mean and sample mean of the Distribution

#Theoretical mean is theomean
theomean
## [1] 5
#Sample mean
hist(means,breaks=50,col="grey",main = "Distribution of 1000 sample means")
abline(v=mean(means),lwd=4,col="red")
text(7,40, paste("Actual mean = ", round(mean(means),4), "\n Theoretical mean = 5" ), col="red")

So we can see that both the mean are very same this is the beauty of CLT

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

The theoretical standard distribution will be “1/lambda/sqrt(n)”

# theoretical standard deviation vs practical standard deviation
print (paste("Theoretical standard deviation: ", round( (1/lambada)/sqrt(sampsize) ,4), ", Practical standard deviation", round(sd(means) ,4) ) )
## [1] "Theoretical standard deviation:  0.7906 , Practical standard deviation 0.7741"
# the variance is
print (paste("Theoretical variance: ", (1/lambada)^2/sampsize, ", Practical variance", round(var(means) ,4) ) )
## [1] "Theoretical variance:  0.625 , Practical variance 0.5992"

So we can say that both are approximetly equal

3.Show that the distribution is approximately normal

We can see it from the histogram itself that it is nearly equal.For clearity I have used the density function.

hist(means,breaks=50,col="orange",main = "Distribution of 1000 sample means",probability = T)
lines(density(means), lwd=3, col="blue")

So, from the graph we can sau that CLT works on exponential distribution too.