---
title: "SDM4 in R: Randomness and Probability (Chapter 15)"
author: "Nicholas Horton (nhorton@amherst.edu) and Sarah McDonald"
date: "June 13, 2018"
output: 
  pdf_document:
    fig_height: 2.8
    fig_width: 6
  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.
require(knitr)
opts_chunk$set(
  tidy = FALSE,     # display code as typed
  size = "small"    # slightly smaller font for code
)
```

## Introduction and background 

This document is intended to help describe how to undertake analyses introduced 
as examples in the Fourth Edition of *Stats: Data and Models* (2014) by De Veaux, Velleman, and Bock.
More information about the book can be found at http://wps.aw.com/aw_deveaux_stats_series.  This
file as well as the associated R Markdown reproducible analysis source file used to create it can be found at http://nhorton.people.amherst.edu/sdm4.

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 15: Randomness and Probability

### Section 15.1: Center (the Expected Value)

We can replicate the calculation on page 390:

```{r}
library(mosaic)
library(readr)
options(digits = 3)
x <- c(10000, 5000, 0)
prob <- c(1/1000, 2/1000, 997/1000)
sum(prob)   # sums to 1
expect <- sum(x*prob)
expect  # expected value
```

### Section 15.2: Spread (The Standard Deviation)

We can continue with the example from page 392:

```{r}
xminmu <- x - expect
xminmu
myvar <- sum(xminmu^2*prob) 
myvar
sd <- sqrt(myvar)
sd
```

### Section 15.3: Shifting and Combining Random Variables

Let's replicate the values from the example on page 394:

```{r}
ex <- 5.83 
varx <- 8.62^2
ed <- ex+5 
ed
vard <- varx
vard
sqrt(vard)
```

### Section 15.5: Continuous random variables

Let's replicate Figure 15.1 (page 400):

```{r}
xpnorm(c(-1, 1), mean=0, sd = 1)
```

and the Think/Show/Tell/Think on pages 402 and 403:

```{r}
sdval <- sqrt(4.50) 
sdval
gf_dist("norm", params = list(18, sdval), xlab = "x", ylab = "f(x)")
xpnorm(20, mean = 18, sd = sdval)  # note how exact value is different from the table!
zval <- (20-18)/sdval
zval
xpnorm(zval, mean = 0, sd = 1)    
```