Statistical Inference Coursera Course Project

Overview:

The objective of this project is to investigate the exponential distribution in R and compare it with the Central Limit Theorem. To generate data for this simulation we will be using lambda = 0.2. The objectives are following: 1. Show the sample mean and compare it to the theoretical mean of the distribution. 2. Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution. 3. Show that the distribution is approximately normal.

Simulations

• We have taken mean of 1000 samples of 40 random exponential numbers with lambda 0.2.

n = 1000 
lambda <- 0.2
mns <- NULL
for(i in 1: n){
        mns <- c(mns, mean(rexp(40, lambda)))
}

Sample mean vs Theoretical mean

• In this section we will calculate theoritical mean of distribution and compare it with sample mean

#Sample Mean
sammean <- mean(mns)
sammean

## [1] 5.017626

#Theoritical Mean
themean <- 1/lambda
themean

## [1] 5

g <- ggplot(data.frame(mns), aes(x = mns)) + 
        geom_histogram(aes(y=..density..), binwidth=.3, colour = "black", alpha = .20) + 
        labs(x = "Mean", y = "Value", title = "Exponential Means Distribution n = 1000")
g + geom_vline(xintercept = sammean, color = "red", size = 1) + labs(title = "Exponential Means Distribution with sample mean")

g + geom_vline(xintercept = themean, color = "blue", size = 1) + labs(title = "Exponential Means Distribution with theoritical mean")

• The simulation mean 5.0176262 is close to theoritical mean 5.

Sample Variance vs Theoretical Variance

• In this section we will calculate theoritical variance of distribution and compare it with sample variance

#Sample standard deviation
samvar <- var(mns)
samvar

## [1] 0.6338725

#Theoritical standard deviation
thevar <- ((1/lambda)/sqrt(40))^2
thevar

## [1] 0.625

• The simulation variance 0.6338725 is close to theoritical variance 0.625.

Distribution

#Plotting the histogram of mean values of 40 random exponential 
g <- ggplot(data.frame(mns), aes(x = mns)) + 
        geom_histogram(aes(y=..density..), binwidth=.3, colour = "black", alpha = .20) + 
        labs(x = "Mean", y = "Value", title = "Distribution of Random Exponential Means") +
        stat_function(fun = dnorm, arg = list(mean = mean(mns), sd = sd(mns)))
g

• The normal density plot shows how the data is approximately Gaussian or normally distributed displaying center limit theorm in the play.
• Also in case we increase number of simulations the data will go towards normal distribution.