ECON 3818

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# ECON 3818
## Chapter 4
### Kyle Butts
### 23 August 2021

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## Chapter 4: Correlation

---
# Multiple Variables
Almost everything we've done so far has been .it[univariate] statistics, but often we're interested in how multiple random variables are related?

- How does education affect earnings?
- How does race affect earnings?
- How does experience affect earnings?

Many events are .it[dependent] on other random variables. In this chapter we'll formalize this concept.

---
# Probability Theory
Recall that with single random variables we characterized probabilities with

- PMF (probability mass function), `$P(X=x)$`, in discrete case
- PDF (probability density function), `$f(x)$`, in continuous case

When we have multiple random variables we use the .hi.purple[joint distribution]

- `$P(X=x, Y=y)$` in the discrete case
- `$f(x,y)$` in the continuous case

The joint distribution describes the probability of obtaining `$X=x$` *and* `$Y=y$`.

- .ex[Example:] If `$X$` is education and `$Y$` is income, you could ask what's the probability that from the population I have someone who earns $65,000/year *and* has a masters degree.

---
# Properties for Joint Distribution
For short hand, `$P(x,y) = P(X=x, Y=y)$`

In this class we'll focus solely on the discrete case

- `$0 \leq P(x,y) \leq 1$`
- `$\sum_x \sum_y P(x,y)=1$`

As long as X and Y are .hi.purple[not independent]
`$$P(x,y) \neq P(x)P(y)$$`

---
# Example
Suppose that `$X$` is the number of girls born out of three kids and `$Y$` is whether the first child is a girl.

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#gpnkvfeqnn .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#gpnkvfeqnn .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#gpnkvfeqnn .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#gpnkvfeqnn .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#gpnkvfeqnn .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#gpnkvfeqnn .gt_left {
  text-align: left;
}

#gpnkvfeqnn .gt_center {
  text-align: center;
}

#gpnkvfeqnn .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#gpnkvfeqnn .gt_font_normal {
  font-weight: normal;
}

#gpnkvfeqnn .gt_font_bold {
  font-weight: bold;
}

#gpnkvfeqnn .gt_font_italic {
  font-style: italic;
}

#gpnkvfeqnn .gt_super {
  font-size: 65%;
}

#gpnkvfeqnn .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span style="display: block; margin-bottom: 8px;">Sample Space</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Outcome</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">$X$</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">$Y$</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">BBB</td>
<td class="gt_row gt_right">0</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">GBB</td>
<td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">BGB</td>
<td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">BBG</td>
<td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">GGB</td>
<td class="gt_row gt_right">2</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">GBG</td>
<td class="gt_row gt_right">2</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">BGG</td>
<td class="gt_row gt_right">2</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">GGG</td>
<td class="gt_row gt_right">3</td>
<td class="gt_row gt_right">1</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Example
Notice that the sample spaces are `$S_X =\{0,1,2,3\}$` and `$S_Y=\{0,1\}$`. The associated joint probabilities are:

<div id="bzsflqrvlf" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#bzsflqrvlf .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#bzsflqrvlf .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#bzsflqrvlf .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#bzsflqrvlf .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#bzsflqrvlf .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#bzsflqrvlf .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#bzsflqrvlf .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#bzsflqrvlf .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#bzsflqrvlf .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#bzsflqrvlf .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#bzsflqrvlf .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#bzsflqrvlf .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
}

#bzsflqrvlf .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#bzsflqrvlf .gt_from_md > :first-child {
  margin-top: 0;
}

#bzsflqrvlf .gt_from_md > :last-child {
  margin-bottom: 0;
}

#bzsflqrvlf .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#bzsflqrvlf .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#bzsflqrvlf .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#bzsflqrvlf .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#bzsflqrvlf .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#bzsflqrvlf .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#bzsflqrvlf .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#bzsflqrvlf .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#bzsflqrvlf .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#bzsflqrvlf .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#bzsflqrvlf .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#bzsflqrvlf .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#bzsflqrvlf .gt_left {
  text-align: left;
}

#bzsflqrvlf .gt_center {
  text-align: center;
}

#bzsflqrvlf .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#bzsflqrvlf .gt_font_normal {
  font-weight: normal;
}

#bzsflqrvlf .gt_font_bold {
  font-weight: bold;
}

#bzsflqrvlf .gt_font_italic {
  font-style: italic;
}

#bzsflqrvlf .gt_super {
  font-size: 65%;
}

#bzsflqrvlf .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span style="display: block; margin-bottom: 8px;">Joint PDF</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">$ Y = 0 $</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">$ Y = 1 $</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">$X = 0 $</td>
<td class="gt_row gt_right">0.125</td>
<td class="gt_row gt_right">0.000</td></tr>
    <tr><td class="gt_row gt_left">$X = 1 $</td>
<td class="gt_row gt_right">0.250</td>
<td class="gt_row gt_right">0.125</td></tr>
    <tr><td class="gt_row gt_left">$X = 2 $</td>
<td class="gt_row gt_right">0.125</td>
<td class="gt_row gt_right">0.250</td></tr>
    <tr><td class="gt_row gt_left">$X = 3 $</td>
<td class="gt_row gt_right">0.000</td>
<td class="gt_row gt_right">0.125</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Example
Let's check this table satisfies the definition of a joint distribution

- `$0 \leq P(x,y) \leq 1 \checkmark$`

- `$\sum_x \sum_y P(x,y)=1$`

`\begin{align}
  \sum_{x \in S_X} \sum_{y \in S_Y} Pr(x,y) &=  Pr(0,0) + Pr(0,1) + Pr(1,0) + Pr(1,1)   \\\\
  &+ Pr(2,0) + Pr(2,1) + Pr(3,0) + Pr(3,1) \\\\
  &= 1/8 + 0 + 2/8 + 1/8 \\\\
  &+ 1/8 + 2/8 + 0 + 1/8 = 1
\end{align}`

---
# Clicker Question
Given the following joint probability mass function, what is the probability of the NASDAQ increasing in value and your portfolio loses value?

<div id="xidrkjwvmz" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#xidrkjwvmz .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#xidrkjwvmz .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#xidrkjwvmz .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#xidrkjwvmz .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#xidrkjwvmz .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#xidrkjwvmz .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#xidrkjwvmz .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#xidrkjwvmz .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#xidrkjwvmz .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#xidrkjwvmz .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#xidrkjwvmz .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#xidrkjwvmz .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
}

#xidrkjwvmz .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#xidrkjwvmz .gt_from_md > :first-child {
  margin-top: 0;
}

#xidrkjwvmz .gt_from_md > :last-child {
  margin-bottom: 0;
}

#xidrkjwvmz .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#xidrkjwvmz .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#xidrkjwvmz .gt_summary_row {
  color: #333333;
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<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span style="display: block; margin-bottom: 8px;">NASDAQ</span></th>
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  </thead>
  <thead class="gt_col_headings">
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  <tbody class="gt_table_body">
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<td class="gt_row gt_right">0.40</td>
<td class="gt_row gt_right">0.05</td></tr>
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<td class="gt_row gt_right">0.15</td>
<td class="gt_row gt_right">0.40</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Clicker Question
Given the following joint probability mass function, what is the probability that the NASDAQ increases in value?

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<td class="gt_row gt_right">0.40</td>
<td class="gt_row gt_right">0.05</td></tr>
    <tr><td class="gt_row gt_left">NASDAQ Decreases</td>
<td class="gt_row gt_right">0.15</td>
<td class="gt_row gt_right">0.40</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Clicker Question
Given the following joint probability mass function, what is the probability that the NASDAQ increases in value, conditional on the portfolio value decreases?

<div id="flqjcwrvsp" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span style="display: block; margin-bottom: 8px;">NASDAQ</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Portfolio Increases</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Portfolio Decreases</th>
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<td class="gt_row gt_right">0.40</td>
<td class="gt_row gt_right">0.05</td></tr>
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<td class="gt_row gt_right">0.15</td>
<td class="gt_row gt_right">0.40</td></tr>
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</table>
</div>

---
# Visualizing a Joint Distribution

The most useful for displaying the relationship between two .ul[quantitative] variables is a .hi.kelly[scatterplot]

- Shows relationship between two quantitative variables
  - Each axis represents a variable
  - Individual data appear as a point, fixed by the values of both variables

---
# Scatterplot Example

---
# Interpreting a Scatterplot

Looking for patterns, and deviations from that pattern
      
- Direction, form, strength of relationship
- Any outliers?
      
Describing the association
      
- .hi.kelly[Positive Association]: above-average values of one tend to accompany above-average values of the other, and below-average values also tend to occur together
- .hi.ruby[Negative Association]: above-average values of one tend to accompany below-average values of the other, and vice versa
      
In general, if one variable is explanatory (influences change) and one is a response variable (outcome), then the explanatory variable is plotted on the x-axis

---
# Correlation
We need to supplement the graph with a numerical measure, generally we use .hi.alice[correlation].

The .alice[correlation] measures the direction and strength of the linear relationship between two quantitive variables. Correlation is usually written as `$r$`

---
# Covariance
In order to understand correlations, we must first discuss .hi.purple[covariance]

Recall: `$V(aX+bY)=a^2V(X)+b^2V(Y)+2ab\cdot cov(X,Y)$`

Covariance measures the joint variability of two random variables

- Sign of covariance explains direction of relationship

- Magnitude of covariance is hard to interpret.sup[*]
  
  - The .alice[correlation coefficient] will not have this problem

- Covariance equals zero whenever X and Y are .hi.purple[independent]

.footnote[.sup[*] Double the units, double the covariance!!]

---
# Covariance

We use the following formula to calculate covariance

`$$cov(X,Y)=E(XY)-E(X)E(Y)$$`

Note: `$E(XY) \neq E(X)E(Y)$` unless X and Y are independent and then cov(X,Y)=0

The magnitude of covariance depends on the units of X and Y

- This means `$cov(A,B) > cov(C,D)$` .hi[does not] imply that A and B have stronger relationship than C and D

- In order to compare relationships we must find a way to normalize their covariances

---
# Correlation

The .hi.alice[correlation] measures the direction and strength of the linear relationship between two quantitive variables. Correlation is usually written as `$r$`

To calculate correlation, we normalize the covariance as so:

$$
  r=\frac{cov(X,Y)}{\sqrt{V(X)}\cdot \sqrt{V(Y)}}
$$

---
# Correlation

.subheader.alice[Notes on correlation]

Values are always between -1 and 1
      
- `$1 \implies$` perfectly linear positive relationship (variables move same direction and same magnitude)

- `$-1 \implies$` pefectly linear negative relationship (variables move in opposite direction but same magnitude) 
      
- Correlations are unit-less

.it.purple[Doesn't imply a causal relationship]

Drawbacks of correlation

- Only measures .it[linear relationships] (we will see what this means)
       
- Just because correlation is zero doesn't necessarily mean variables are independent
      
- Not resistant to outliers

---
# Correlations - Example

---
# Correlations - Example

---
# Correlations - Visualized

---
# Why Correlation isn't Perfect

Bottom row is an example of non-linear relationships

<!-- 
--- 
class: center, middle

![The Datasaurus Dozen](data:image/png;base64,#dino.gif) 
-->

---
# Covariance and Independence
Since covariance (and correlations) only measure linear relationships:

`$$cov(X,Y) = 0 \not\implies X \text{ and } Y \text{ are independent}$$`

<br/>
However, since `$E(XY)=E(X)E(Y)$` when X \text{ and } Y are independent:

`$$X \text{ and } Y \text{ are independent} \rightarrow cov(X,Y)=0$$`

---
# Joint Distributions
When calculating the covariance we use equation `$cov(X,Y) = E(XY)-E(X)E(Y)$`

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`$$E(XY)=x\cdot y \cdot P(x,y)$$`
In this example:

`$$E(XY)=(0\cdot 0 \cdot 1/8) + (0\cdot  1\cdot 0) + (1\cdot 0 \cdot 2/8) + (1\cdot 1 \cdot 1/8) +$$`
`$$(2\cdot 0 \cdot 1/8) + (2\cdot 1 \cdot 2/8) + (3\cdot 0 \cdot 0) + (3\cdot 1 \cdot 1/8)=1$$`

---
# Marginal Probabilties
In order to calculate `$E(X)$` and `$E(Y)$` from a joint distribution we must first calculate the .hi.purple[marginal probabilities] of both X and Y.

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<td class="gt_row gt_left">1/8</td>
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<td class="gt_row gt_left">2/8</td>
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<td class="gt_row gt_left">1/8</td>
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<td class="gt_row gt_left">4/8</td>
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These marginal probabilities, `$P(X=x)$` are calculated adding up the probabilities across each scenario where `$X=x$`

---
# Marginal Probabilities
We can use these marginal probabilities to calculate `$E(X)$` and `$E(Y)$`.

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<td class="gt_row gt_left">1/8</td>
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<td class="gt_row gt_left">1/8</td></tr>
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<td class="gt_row gt_left">2/8</td>
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<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">2/8</td>
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<td class="gt_row gt_left">0</td>
<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">1/8</td></tr>
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<td class="gt_row gt_left">4/8</td>
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</div>

`$E(X)=(0 \cdot 1/8) + (1 \cdot 3/8) + (2 \cdot 3/8) + (3 \cdot 1/8) =1.5$`
`$E(Y)=(0 \cdot 4/8) + (1 \cdot 4/8) =0.5$`

---
# Covariance of Joint Distribution
All of that work leads us here:

`$$E(XY)=1$$`
$$E(X)= 1.5 $$
`$$E(Y)= 0.5$$`
`$$cov(X,Y)=E(XY)-E(X)E(Y) =1 - (1.5 \cdot 0.5) = 0.25$$`

What does `$0.25$` mean? Is this a strong relationship? A weak one?

---
# Covariance to Correlation

Again, we often use correlation instead of covariance because correlation .hi[does not depend on the units]

To find correlation from covariance we use the following equation:

`$$r = \frac{\text{cov(X,Y)}}{\sqrt{\text{V(X)}\cdot \text{V(Y)}}}$$`
So we need to calculate the variance of X and Y, using information about the joint probabilities

---
# Covariance to Correlation
Recall the joint probabilities we gathered from the table

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  padding-right: 5px;
  overflow-x: hidden;
}

#rgjchmavre .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#rgjchmavre .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#rgjchmavre .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#rgjchmavre .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#rgjchmavre .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
}

#rgjchmavre .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#rgjchmavre .gt_from_md > :first-child {
  margin-top: 0;
}

#rgjchmavre .gt_from_md > :last-child {
  margin-bottom: 0;
}

#rgjchmavre .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
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  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#rgjchmavre .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
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  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#rgjchmavre .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#rgjchmavre .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#rgjchmavre .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#rgjchmavre .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#rgjchmavre .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
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#rgjchmavre .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
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  border-bottom-color: #D3D3D3;
}

#rgjchmavre .gt_footnotes {
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  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#rgjchmavre .gt_footnote {
  margin: 0px;
  font-size: 90%;
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#rgjchmavre .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
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  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#rgjchmavre .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#rgjchmavre .gt_left {
  text-align: left;
}

#rgjchmavre .gt_center {
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#rgjchmavre .gt_font_normal {
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#rgjchmavre .gt_font_bold {
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#rgjchmavre .gt_font_italic {
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  font-weight: normal;
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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">$Y = 0$</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">$Y = 1$</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">$ Pr(X) $</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">$X = 0$</td>
<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">0</td>
<td class="gt_row gt_left">1/8</td></tr>
    <tr><td class="gt_row gt_left">$X = 1$</td>
<td class="gt_row gt_left">2/8</td>
<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">3/8</td></tr>
    <tr><td class="gt_row gt_left">$X = 2$</td>
<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">2/8</td>
<td class="gt_row gt_left">3/8</td></tr>
    <tr><td class="gt_row gt_left">$X = 3$</td>
<td class="gt_row gt_left">0</td>
<td class="gt_row gt_left">1/8</td>
<td class="gt_row gt_left">1/8</td></tr>
    <tr><td class="gt_row gt_left">$Pr(Y)$</td>
<td class="gt_row gt_left">4/8</td>
<td class="gt_row gt_left">4/8</td>
<td class="gt_row gt_left">1</td></tr>
  </tbody>
  
  
</table>
</div>

$$ E(X^2) = (0^2 \cdot 1/8) + (1^2 \cdot 3/8) + (2^2 \cdot 3/8) + (3^2 \cdot 1/8) = 3$$

$$ E(Y^2)=(0^2 \cdot 4/8) + (1^2 \cdot 4/8) = 0.5$$

---
# Covariance to Correlation
`$E(X)=1.5$` and `$E(X^2)=3 \rightarrow V(X)=3-1.5^2= 0.75$`

`$E(Y)=0.5$` and `$E(Y^2)=0.5 \rightarrow V(Y)=0.5-0.5^2 = 0.25$`

`$cov(X,Y)=0.25$`

`$$r = \frac{\text{cov(X,Y)}}{\sqrt{\text{V(X)}}\cdot \sqrt{\text{V(Y)}}} =\frac{0.25}{\sqrt{0.75}\cdot\sqrt{0.25}}=0.577$$`

---
# Clicker Question
What can be said of the correlation between the brand of an automobile and its quality?

<ol type = "a">
  <li>The correlation is negative, because smaller cars tend to have higher quality and larger cars tend to have lower quality.</li>
  <li>The correlation is positive, because better brands have higher quality.</li>
  <li>If the correlation is negative, an arithmetic mistake was made; correlation must be positive.</li>
  <li>Correlation makes no sense here, because brand is a categorical variable.</li>
</ol>

---
# Clicker Question
Which of the following statements is false?
<ol type = "a">
  <li>Older men tend to have lower muscle density, so the correlation between age and muscle density in older men must be negative.</li>
  <li>Older children tend to be taller than younger children, so the correlation between age and height in children must be positive.</li>
  <li>A researcher finds that the correlation between two variables is close to 0, so the two variables must be unrelated.</li>
  <li>Taller people tend to be heavier than shorter people, so the correlation between height and weight must be positive.</li>
</ol>