Statistical Interference Course Project 1:

analysis of normal distribution

by Henrik Gjerning, 27th September 2015

A simulation will be used to explore inference and do some simple inferential data analysis.

This project investiges the exponential distribution in R and compares it with the central Limit Theorem (CLT). The project consists of two parts:

            1. Setting up simulation parameters and calc of statistics
            2. Sample Mean vs Theoretical Mean
            3. Sample Variance vs Theoretical Variance
            4. Distribution, is it normal?

The analysis will compare samples to theorecital distributions to evaluate their similarities based on the parameteres below.

For the analysis we use the following Input:

           a) Lambda:          0.2
           b) n:                40
           c) simulations:    1000

1. Setting up simulation parameters and calc of statistics

set.seed(2)
lambda <- 0.2
n <- 40
simulations <- 1000

# simulate 
simulated_exponentials <- replicate (simulations, rexp(n, lambda))

# calculate mean of exponentials
means_exponentials <- apply (simulated_exponentials, 2, mean)

actual_mean <- mean (means_exponentials)
theory_mean <- 1/lambda
theory_sd <- ((1/lambda) * (1/sqrt(n)))

actual_sd <- sd(means_exponentials)
theory_var <- theory_sd^2
actual_var <- var(means_exponentials)

2. Sample Mean vs Theoretical Mean

hist(means_exponentials, col="blue", main="Theoretical vs Actual Mean", breaks=20)
abline(v=mean(actual_mean), lwd="4", col="red")
text(6, 90, paste("Actual mean = ", round(mean(actual_mean),2), "\n Theoretical Mean = 5" ), col="red")

### Results:
cat("The theoretical mean is defined as:\n 1/lambda, giving 1/0.2 = 5 \n whereas the simulated mean is: ",actual_mean) +
cat("\n Giving a difference of: ", 5 - actual_mean)

## The theoretical mean is defined as:
##  1/lambda, giving 1/0.2 = 5 
##  whereas the simulated mean is:  5.016356
##  Giving a difference of:  -0.01635615

## numeric(0)

Conclusion: So the number looks quite similar

3. Sample Variance vs Theoretical Variance

cat(" the theoretical variance is defined as: \n 1/lambda / sqrt(n)^2, giving 1/0.2 / sqrt(40)^2 = 0.625 \n whereas the simulated variance is: ",actual_var) +
cat("\n Giving a difference of: ", 0.7906 - actual_var)

##  the theoretical variance is defined as: 
##  1/lambda / sqrt(n)^2, giving 1/0.2 / sqrt(40)^2 = 0.625 
##  whereas the simulated variance is:  0.6691305
##  Giving a difference of:  0.1214695

## numeric(0)

Conclusion: So the number looks quite similar

4. Distribution, is it normal?

hist(means_exponentials, prob=TRUE, col="green", main="Mean distribution for simulation", breaks=20)
lines(density(means_exponentials), lwd=3, col="darkgreen")

Conclusion: So the distribution looks normal (gaussian) distributed