exponential_distribution.Rmd

---
output: 
  pdf_document:
    keep_tex: true
    fig_caption: true
    latex_engine: pdflatex
title: "Exponential Distribution and Central Limit Theorem"
abstract: "***This document provides the assignment 'Course Project Part 1' for Coursera's Statistical Inference Class in the Coursera Data Science series. Replication files are available on the author's Github account (https://github.com/tomfischersz).***"
author: "Thomas Fischer"
date: "`r format(Sys.time(), '%B %d, %Y')`"
geometry: margin=1in
linkcolor: "blue"
---

```{r setup, include=TRUE, echo=FALSE}
knitr::opts_chunk$set(fig.pos= "h", out.extra = '')
```


```{r req_libraries, echo=FALSE, message=FALSE}
require(ggplot2)
```
```{r ref.label="simulation", echo=FALSE, eval=TRUE}
```
```{r ref.label="calc_mean", echo=FALSE, eval=TRUE}
```
```{r ref.label="calc_var", echo=FALSE, eval=TRUE}
```
```{r ref.label="fig_hist_1", echo=FALSE, eval=TRUE}
```

# 1. Synopsis

The aim of this report is to investigate the sampling distribution for $\bar X_n$, derived as the mean from samples with size $n$ from an exponential distribution. The Central Limit Theorem (CLT) states, that for large $n$ we should except our sampling distribution to be approximately normal with mean $\mu_{\bar X} =\mu$ and standard deviation $\sigma_{\bar X} =$$\frac{\sigma}{\sqrt{n}}$ (also called standard error of the mean).

# 2. Simulation

The first step is to simulate 1000 random variables each calculated as the mean of $n = 40$ random exponential variables with population parameter $\lambda = 0.2$ (rate). To generate random exponential variables we use the R function rexp(). The resulting 1000 sample means, which themselves are random variables, are stored in a vector ([Code](#Appendix_1)).


# 3. Sample Mean versus Theoretical Mean

The theoretical mean is the population mean of our exponential distribution with $\lambda = 0.2$ and is given as $\mu = \frac{1}{\lambda}$. According to the Law of Large Numbers we also know that the sampling distribution for $\bar X_n$ is centered at $\mu$, therefore $\mu_{\bar X} = \frac{1}{\lambda}$. We can now compare this theoretical mean with the actual mean of the simulation of 1000 means from samples of size 40 ([Code](#Appendix_2)). The following table shows, that the two calculated values are very close to each other:  

|Theoretical Mean|Sample Mean|Deviation from Theoretical|
|--|--|--|
|`r theoretical_mu`|`r mu_sample_means`|`r dev_mean`|  

# 4. Sample Variance versus Theoretical Variance

We now have a closer look at the variability or spread of our sampling distribution of means. The theoretical variance of the mean of samples with size $n$ from iid random exponential variables is given as $VAR(\bar X) = \frac{\sigma^2}{n}$, where $\sigma^2$ is the variance of the population we sampled from which is $\frac{1}{\lambda^2}$. We therefore can rewrite $VAR(\bar X) = \frac{1}{\lambda^2 n}$. We calculate the theoretical variance and the actual variance ([Code](#Appendix_4)) and show it as a table:  

|Theoretical Variance|Actual Variance|Deviation from Theoretical|
|--|--|--|
|`r theoretical_var`|`r var_sample_means`|`r dev_variance`|  

# 5. Distribution

Finally we want to examine if the distribution of our simulation of 1000 averages of 40 random exponentials is approximately normal. In figure \ref{fig:fig_hist_1} we plot the histogram of our simulation data together with the theoretical normal distribution we would except according to the CLT. In figure \ref{fig:fig_qqnorm} we plot a Q-Q (Quantile-Quantile) Plot, where we compare the theoretical quantiles with observed quantiles.

```{r plot_hist_1, echo=FALSE,fig.height=2.8,fig.width=5.5, fig.show='hold',fig.align='center',fig.cap="\\label{fig:fig_hist_1}Sampling distribution of the sample mean. The red line shows the standard normal pdf we would except according the CLT."}
plot(hist_1)
```

```{r ref.label="fig_qqnorm", echo=FALSE,eval=TRUE,fig.height=2.8,fig.width=5.5, fig.show='asis',fig.align='center',fig.cap="\\label{fig:fig_qqnorm}Quantile-Quantile Plot of the observed sampling distribution of the sample mean versus a normal distribution with mean = 5 and standard deviation = 0.79."}
```

# 6. Conclusion

The histogram of our sample means in figure \ref{fig:fig_hist_1} shows a density distribution that is quite normal and the Q-Q Plot in figure \ref{fig:fig_qqnorm} shows a nearly linear plot. That let us conclude that in fact our simulated distribution of 1000 averages of 40 random exponential variables follows quite good a normal gaussian distribution ${N}(5, \frac{5}{\sqrt{40}})$. That is what we expected from the Central Limit Theorem.  

\newpage

# Appendix: R Source Code

### 1. Code for creating our simulation: {#Appendix_1}
```{r simulation, eval=FALSE}
lambda <- 0.2 # rate parameter
n <- 40 # sample size
no_simulations <- 1000 # number of samples
set.seed(1347)
sample_means <- replicate(n = no_simulations, mean(rexp(n, rate = lambda)))
```

### 2. Calculating the theoretical mean and the mean of the 1000 sample means: {#Appendix_2}
```{r calc_mean, eval=FALSE}
theoretical_mu <- 1/lambda
mu_sample_means <- mean(sample_means)
dev_mean <- paste(round((mu_sample_means - theoretical_mu) /
                            theoretical_mu * 100, 2),'%')
```


### 3. Calculating theoretical and actual variance: {#Appendix_4}
```{r calc_var, eval=FALSE}
theoretical_var <- 1/(lambda^2 * n)
var_sample_means <- var(sample_means)
dev_variance <- paste(round((var_sample_means - theoretical_var) /
                                theoretical_var * 100, 2),'%')
```

### 4. Histogram 1: {#Appendix_5}
```{r fig_hist_1, eval=FALSE}
hist_1 <- ggplot(data = data.frame(sample_means), aes(x=sample_means))
hist_1 <- hist_1 + geom_histogram(color = 'black', binwidth = 0.2,
                                  fill = '#51B5E0', aes(y=..density..))
hist_1 <- hist_1 + labs(x = "Sample Means",
                        y = "Density",
                        title = "Density Histogram of Sample Means")
hist_1 <- hist_1 + stat_function(fun = dnorm,
                                 args = list(mean = theoretical_mu,
                                             sd = sqrt(theoretical_var)),
                                 colour = "red", size=1)
```

### 5. Q-Q Plot
```{r fig_qqnorm, eval=FALSE}
qqnorm(sample_means, main = "Normal Q-Q Plot")
qqline(sample_means)
```