---
title: "STAT 491 - Lecture 4"
date: January 23, 2018
output: pdf_document
---

```{r setup, include=FALSE}
library(knitr)
knitr::opts_chunk$set(echo = TRUE)
set.seed(01162018)
```

# Ch.3 Probability - continued

### Normal probability density function
The most common distribution is the normal or (Gaussian) distribution which is associated with the bell-curve.
\vfill

```{r, out.width = '400px',fig.align='center', echo = F}
x <- seq(-3, 3, by = .1)
plot(x, dnorm(x), type='l', main = 'Standard Normal Density', ylab = 'p(x)')
```

\vfill
The density function of a normal distribution can be written as:

\vfill 
\vfill
where:

- 
\vfill
- 
\vfill

Note that the density of the distribution ($p(x)$) can be greater than one, let $x = \mu$. 
\vfill

\vfill
\newpage

### Mean and Variance of the distribution

- *QUESTION:* Consider the distribution generated by rolling a die, what is the *Mean* of the distribution and how to we interpret it?:
\vfill
- _answer_: 
\vfill
- *QUESTION:* Show mathematically the result above about the mean of the die rolls:
\vfill
- _answer_: 

\vfill
- Expectation with a continuous distribution. Now assume that $x$ is a continuous quantity such as height. Now there are a countably infinite number of possibilities for $x$, how do we rewrite the expectation formula?
\vfill

\newpage
- Consider an occupancy model, where the goal is to determine whether a species inhabits a specific region. We will later see that this translates to one of those coin flip models. As a starting point, you have no information to inform your prior belief on the probability that bats reside at a specific location, so you use a uniform model

```{r, echo=F}
x <- seq(0, 1, by = .01)
plot(x, dunif(x), type = 'l', ylab = 'p(x)', main='Uniform Density')
```

1. Write the density of the uniform distribution:

\vfill

2. Compute the expected value of this distribution:

\vfill
- We previously talked about the standard deviation in the context of the normal distribution, how does that relate to _variance_? 
\vfill
- The variance is computed as:
\vfill
- We will do little calculation of variance in this class, rather we will typically use known properties from distributions.

\newpage

### Highest Density Interval (HDI)
- Rather than summarizing a distribution with the mean and variance, we might also use intervals. To motivate this idea, consider the two distributions below and guess the mean and variance.

```{r,fig.align='center', echo=F, out.width='400px'}
x <- seq(-10,10, by = .01)
par(mfcol=c(1,2))
plot(x, dnorm(x, mean = 2, sd = sqrt(4) ), type='l', ylab='p(x)', ylim=c(0,.5))
shape.val <- 1; scale.val <- 2
plot(x, dgamma(x, shape = shape.val, scale = scale.val ), type='l', ylab='p(x)', ylim=c(0,.5))
par(mfcol=c(1,1))
```

- \vfill

- Instead we will use an interval to describe them, (although the distribution as a whole is quite telling in this setting).

- The Highest Density Interval (HDI), \vfill

- Include a rough sketch of the 95% HDI on the figures below, note $p(0)=.5$ and $p(x) = 0$ for $x < 0$ on the figure on the right :    
```{r,fig.align='center', echo=F, out.width='400px'}
x <- seq(-10,10, by = .01)
par(mfcol=c(1,2))
plot(x, dnorm(x, mean = 2, sd = sqrt(4) ), type='l', ylab='p(x)', ylim=c(0,.5))
#abline(v = qnorm(.025, mean = 2, sd = sqrt(4)), col='gray',lwd=2, lty=3)
#abline(v = qnorm(.975, mean = 2, sd = sqrt(4)), col='gray',lwd=2, lty=3)
shape.val <- 1; scale.val <- 2
plot(x, dgamma(x, shape = shape.val, scale = scale.val ), type='l', ylab='p(x)', ylim=c(0,.5))
#abline(v = qgamma(.95, shape = shape.val, scale = scale.val), col='gray',lwd=2, lty=3)
#abline(v = 0, col='gray',lwd=2, lty=3)
par(mfcol=c(1,1))
```

\newpage

## Two-Way Distributions
What we have seen thus far have been primarily univariate distribution. Now consider multivariate distributions such as:

- 
\vfill

Or consider the table below (from the textbook) that contains hair color and eye color:

|                       |       | Hair Color |      |        |                      |
|:---------------------:|-------|:----------:|------|:------:|----------------------|
| Eye Color             | Black | Brunette   | Red  | Blond | Marginal (eye color) |
| Brown                 | 0.11  | 0.20       | 0.04 | 0.01   | 0.37                 |
| Blue                  | 0.03  | 0.14       | 0.03 | 0.16   | 0.36                 |
| Hazel                 | 0.03  | 0.09       | 0.02 | 0.02   | 0.16                 |
| Green                 | 0.01  | 0.05       | 0.02 | 0.03   | 0.11                 |
| Marginal (hair color) | 0.18  | 0.48       | 0.12 | 0.21   | 1.0                  |
 
and calculate the following quantities:

- $p(\text{h} = \text{blond})$ =  \vfill
- $p(\text{e} = \text{blue})$ = \vfill
- $p(\text{h} = \text{blond}, \text{e} = \text{blue})$ =  \vfill
- $p(\text{h} = \text{blond}) = $,
this is known as a _marginal_ distribution that is obtained my marginalizing or integrating out the other variable. \vfill
- Formally, the _marginal_ distribution can be defined as:
    - \vfill
    - \vfill
    
\newpage

### Conditional Probability

With multivariate distributions, researchers are often interested in the distribution of one variable given a specific value of another variable.  For instance consider,

- \vfill
- \vfill
\vfill

This is known as a conditional probability and will be an essential element in Bayesian data analysis. \vfill

- *QUESTION:* using the hair - eye color table, answer the following questions:

    - What is the probability of a person having red hair:    \vfill
    - What is the probability of a person having blue eyes:  \vfill
    - What is the probability of a person having blue eyes and red hair:  \vfill
    - What is the probability of a person having blue eyes given that they have red hair: \vfill
    - What is the probability of a person having red hair given that they have blue eyes: 
    \vfill
Thus far the quetions were answered with intuition, how can we mathematically formulate conditional probability for two events, $A$ and $B$?
\vfill

\vfill


- This can also be sketched out using a Venn diagram.
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\newpage

### Independence of Attributes

Recall the two events we stated previously:

- \vfill
- \vfill
Do you anticipate that these probabilities  are the same?
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Mathematically, two events ($A$ and $B$) are _independent_ if : 
\vfill
In otherwords, if knowing $B$ give you no additional insight on the distribution of $A$, then $A$ and $B$ are independent.\vfill
Two events are also independent if 
\vfill
 - Are having blue eyes and red hair independent events?
 \vfill