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---

title: 'Statistical Inference Course Project Part 1: Simulation'

author: "Massimiliano Figini"

date: "October 9, 2016"

output: pdf_document

geometry: margin=2cm

---

Investigation of the exponential distribution in R and comparation with the Central Limit Theorem.

Instruction:

1) 1000 Simulation

2) 40 exponentials

3) Lambda = 0.2

```{r p1}

# Basic settings

knitr::opts_chunk$set(echo = TRUE,tidy.opts=list(width.cutoff=60),tidy=TRUE)

set.seed(1983)

# Main variables

Lambda = 0.2

Simulation <- NULL

for (i in 1 : 1000) Simulation = c(Simulation, mean(rexp(40,Lambda)))

```

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

Theoretical mean is 1/Lambda, sample mean is the mean of the simulation values.

```{r p1_q1}

TheoreticalMean <- 1/Lambda

SampleMean <- mean(Simulation)

paste("Theoretical mean is",round(TheoreticalMean,3),"sample mean is",round(SampleMean, 3))

```

The sample mean is very close to the theoretical mean.

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

I can calculate theoretical variance by dividing 1/Lambda for the square root of the number of exponenentials and squaring all.

```{r p1_q2}

TheoreticalVariance <- ((1/Lambda)/sqrt(40))^2

SampleVariance <- var(Simulation)

paste("Theoretical variance is",TheoreticalVariance,"sample variance is",round(SampleVariance, 3))

```

The sample variance is very close to the theoretical variance.

## QUESTION 3: Show that the distribution is approximately normal.

Using an histogram I can easily see if the distribution is approximately normal.

```{r p1_q3}

hist(Simulation, probability = TRUE,breaks = 40, col="red", main="Distribution", xlab="Simulation means")

lines(density(Simulation), lwd=3, col="black")

```

The distribution of averages of random sampled exponentials is like a normal distribution.

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