project1.Rmd

---
title: "Statistical Inference Project Part 1"
author: "Moisés R. Santos"
#date: "05/05/2020"
output:
  pdf_document: default
  html_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Purpose

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


```{r}

set.seed(42) # seed for reproductibility
lambda <- 0.2
n <- 40 # number of exponentials
sim <- 1000 # number of simulations

mns = NULL
for (i in 1 : 1000) mns = c(mns, mean(rexp(n, lambda)))
hist(mns)

```

As can be seen in the histogram, this distribution is close to normal distribution.

## Sample Mean versus Theoretical Mean

The mean $\mu$ of a exponential distribution of rate $\lambda$ is $\mu = \frac{1}{\lambda}$, therefore


```{r}
mu = 1/lambda
mu
```

Since the sample mean of the 1000 simulations, we have shown that expected mean is close to sample mean.

```{r}
mean(mns)
```


## Sample Variance versus Theoretical Variance

The variance $Var$ of a exponential distribution of rate $\lambda$ is $Var = \sigma^2 = \frac{\mu^2}{n}$, therefore


```{r}
Var = mu^2/n
Var
```

Since the sample variance of the 1000 simulations, we have shown that expected variance is close to sample variance.

```{r}
var(mns)
```



## Distribution

Comparison of a normal distribution with mean $\mu= \lambda$ and standard deviation = $ \frac{\mu}{\sqrt{n}}$ with the simulations density plot. In green, we have the normal distribution and in red the density of the simulations histogram.


```{r}
hist(mns, freq = FALSE, col="blue", xlab = "Simulation Mean", ylab = "Frequency", main = "Histogram of distribution")
curve(dnorm(x, mean=lambda^-1, sd=(lambda*sqrt(n))^-1), add=TRUE, col='green', lwd=2)
lines(density(mns),col="red",lwd=2)
```

As a result, we have that the density of the simulations approaches the normal curve with the theoretical values.