---
title: "IPS9 in R: Probability: The Study of Randomness (Chapter 4)"
author: "Bonnie Lin and Nicholas Horton (nhorton@amherst.edu)"
date: "July 26, 2018"
output: 
  pdf_document:
    fig_height: 3
    fig_width: 5
  html_document:
    fig_height: 3
    fig_width: 5
  word_document:
    fig_height: 4
    fig_width: 6
---


```{r, include = FALSE}
# Don't delete this chunk if you are using the mosaic package
# This loads the mosaic and dplyr packages
require(mosaic)
```

```{r, include = FALSE}
# knitr settings to control how R chunks work.
knitr::opts_chunk$set(
  tidy = FALSE,     # display code as typed
  size = "small"    # slightly smaller font for code
)
```

## Introduction and background 

These documents are intended to help describe how to undertake analyses introduced 
as examples in the Ninth Edition of \emph{Introduction to the Practice of Statistics} (2017) by Moore, McCabe, and Craig.

More information about the book can be found [here](https://macmillanlearning.com/Catalog/product/introductiontothepracticeofstatistics-ninthedition-moore).
The data used in these documents can be found under Data Sets in the [Student Site](https://www.macmillanlearning.com/catalog/studentresources/ips9e?_ga=2.29224888.526668012.1531487989-1209447309.1529940008#). This
file as well as the associated R Markdown reproducible analysis source file used to create it can be found at https://nhorton.people.amherst.edu/ips9/.

This work leverages initiatives undertaken by Project MOSAIC (http://www.mosaic-web.org), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the `mosaic` package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignettes (http://cran.r-project.org/web/packages/mosaic).  A paper describing the mosaic approach was published in the *R Journal*: https://journal.r-project.org/archive/2017/RJ-2017-024.
  
## Chapter 4: Probability: The Study of Randomness
This file replicates the analyses from Chapter 4: Probability: The Study of Randomness.

First, load the package that will be needed for this document: 
```{r load-package}
library(mosaic)
```

### Section 4.1: Randomness
The results of two trials of a coin toss simulation 5000 times
are plotted as the proportion of heads, as shown in Figure 3.1 (page 216). 
We can emulate one trial of such simulation as a plot by typing: 
```{r eg4-1}
tosses <- rbinom(n = 1:5000, size = 1, prob = 0.5)
x <- seq(1:5000)
cy <- cumsum(tosses)
phead <- (cy / x)
gf_line(phead ~ x, color = "red") %>% 
  gf_labs(x = "Number of tosses", y = "Proportion of heads") %>% 
  gf_hline(., yintercept = 0.5)
```

The parameters in the `rbinom()` function can be explained in the following way: 
n = 1:5000 specifies the number of observations, while size = 1 specifies the number 
of trials, with each trial getting a probability of success equal to 0.5. 
Another way to think about this is: Draw either 0 or 1, given there is a 50% chance of 
selecting either number, 5000 times.  

Another useful function that is related to this xpbinom(), which can be used to plot the proab

The R function, `runif()`, generates a random number between 0 and 1. 
We can demonstrate using the code below: 
```{r eg4-7}
runif(1)
```
Since the default arguments in the function define the sample space to be all numbers 
between 0 and 1, all we need to specify the number of random numbers we want outputted.
Run the code above several times. Notice that every iteration gives you a different output. 
If you do not set a seed, every time you run the code, you will get a random number. 

To demonstrate adjustments that you can make to the `runif()` call: 
```{r}
runif(2) #change the number of random numbers generated from this sample space
set.seed(2018) #setting the seed to get reproducible results
runif(1)
```
Now that we have selected a pseudorandom seed, anyone who runs this code should expect to see
the output of `runif(1)` to be 0.34. 

### Section 4.2: Probability models

### Section 4.3: Random variables
We can also display probability histograms that compare the probability model for equally likely random digits with the model given by Benford's law (page 237): 
```{r use-your-knowledge-4-42}
# Figure 4.5 (a)
eq_likely <- data.frame(Outcomes = rbinom(1000, 1:9, 0.111))
# Figure 4.5 (b)
benlaw <- c(0.301, 0.176, 0.125, 0.097, 0.079, 0.067, 0.058, 0.051, 0.046)
digits <- 1:9
bendata <- data.frame(prob = benlaw, x = digits)
gf_point(prob ~ x, data = bendata)
```

Below is the code to generate a probability histogram for the distribution of 
the number of heads in 2000 trials of tossing a coin four times as shown in 
Figure 4.7 (page 239): 
```{r heads}
trials <- data.frame(Heads = rbinom(2000, size = 4, prob = 0.5))
gf_histogram(~ as.factor(Heads), data = trials, stat = "count") %>% 
  gf_labs(x = "Number of Heads")
```

### Section 4.4: Means and variances of random variables

### Section 4.5: General probability rules