ECON 3818

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# ECON 3818
## Distributions
### Kyle Butts
### 10 August 2021

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## Distributions

---
# Outline

Discrete Case

- Probability Mass Function
- Calculating probabilities

Continuous Case

- Probability Density Function
- Calculating probabilities

---
# Probability Distribution Functions

Until now we have discussed probability distributions in very loose terms. We will build a formal definition of a .hi.cranberry[probability distribution function].

First we consider the discrete case, and then the continuous case.

---
class: clear,middle

### Discrete

---
# Defining Probability Mass Function
Let `$X$` be a discrete random variable defined over sample space `$S$` with outcome `$x \in S$`.

The .hi.cranberry[probability mass function] (or pmf) of `$X$` is a function that assigns a probability value to every possible outcome of `$X$`.

We write this as `$P_X(x)$`, probability of `$X=x$`.

---
# Example PMF
.subheader.coral[Explicitly Given]

`$X$` is defined as the number of people seated at a random table at a restaurant. The PMF of X is provided below:

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  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

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

#vcmfmmgkwb .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;
}

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

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

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

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

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

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

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

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

#vcmfmmgkwb .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="2" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Probability Distribution of X</th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">x</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P(X = x)</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">1</td>
<td class="gt_row gt_right">0.07</td></tr>
    <tr><td class="gt_row gt_left">2</td>
<td class="gt_row gt_right">0.36</td></tr>
    <tr><td class="gt_row gt_left">3</td>
<td class="gt_row gt_right">0.32</td></tr>
    <tr><td class="gt_row gt_left">4</td>
<td class="gt_row gt_right">0.21</td></tr>
    <tr><td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">5 or more</td>
<td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">0.04</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Example PMF
.subheader.coral[Based on Scenario]

Suppose you flip a fair coin twice. Let `$X$` be the number of heads that appear. The pmf of `$X$` is

<div id="jpuvyrcvpr" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Chivo: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: Chivo, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
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#jpuvyrcvpr .gt_table {
  display: table;
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  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;
}

#jpuvyrcvpr .gt_heading {
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  text-align: left;
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  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
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#jpuvyrcvpr .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
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  font-weight: initial;
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  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
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#jpuvyrcvpr .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
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#jpuvyrcvpr .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;
}

#jpuvyrcvpr .gt_col_heading {
  color: #333333;
  background-color: white;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
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  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;
}

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

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

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

#jpuvyrcvpr .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%;
}

#jpuvyrcvpr .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;
}

#jpuvyrcvpr .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;
}

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

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

#jpuvyrcvpr .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;
}

#jpuvyrcvpr .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;
}

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

#jpuvyrcvpr .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;
}

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

#jpuvyrcvpr .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;
}

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

#jpuvyrcvpr .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;
}

#jpuvyrcvpr .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;
}

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

#jpuvyrcvpr .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;
}

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

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

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

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

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

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

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

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

#jpuvyrcvpr .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="2" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Probability Distribution of X</th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">x</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P_X(x)</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">0</td>
<td class="gt_row gt_right">0.25</td></tr>
    <tr><td class="gt_row gt_left">1</td>
<td class="gt_row gt_right">0.50</td></tr>
    <tr><td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">2</td>
<td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">0.25</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Example PMF

---
# Properties of PMFs

We say a `$p_X(x)$` is a .hi.purple[valid] pmf if it satisfies the following:

1. `$0 \leq p_X(x) \leq 1$` for all `$x \in S$`.

2. `$\displaystyle\sum_{x \in S} p_X(x) = 1.$`

---
# Using PMFs

We can use the PMF to answer questions about cumulative probabilities, for example:
Recall the previous example:

<div id="usvxzrouqy" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Chivo: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: Chivo, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#usvxzrouqy .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;
}

#usvxzrouqy .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;
}

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

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

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

#usvxzrouqy .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;
}

#usvxzrouqy .gt_col_heading {
  color: #333333;
  background-color: white;
  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;
}

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

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

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

#usvxzrouqy .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%;
}

#usvxzrouqy .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;
}

#usvxzrouqy .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;
}

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

#usvxzrouqy .gt_from_md > :last-child {
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}

#usvxzrouqy .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;
}

#usvxzrouqy .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;
}

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

#usvxzrouqy .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;
}

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

#usvxzrouqy .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;
}

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

#usvxzrouqy .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;
}

#usvxzrouqy .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;
}

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

#usvxzrouqy .gt_sourcenotes {
  color: #333333;
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<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="2" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Probability Distribution of X</th>
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  </thead>
  <thead class="gt_col_headings">
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    <tr><td class="gt_row gt_left">1</td>
<td class="gt_row gt_right">0.07</td></tr>
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<td class="gt_row gt_right">0.36</td></tr>
    <tr><td class="gt_row gt_left">3</td>
<td class="gt_row gt_right">0.32</td></tr>
    <tr><td class="gt_row gt_left">4</td>
<td class="gt_row gt_right">0.21</td></tr>
    <tr><td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">5 or more</td>
<td class="gt_row gt_right" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">0.04</td></tr>
  </tbody>
  
  
</table>
</div>

What is the probability a random table at the restaurant has 2 or 3 people seated?
$$ 
P(X=2) = 0.36 \text{ and } P(X=3) = 0.32 \implies 
$$

$$
P(X=2 \text{ or } 3) = 0.36 + 0.32 = .68 
$$

---
# Clicker Question
Assume there are four outcomes of `$X$`: `$1, 5, 10$` and `$20$`. Given the following PMF, what is the probability X=20?

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<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="2" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Probability Distribution of X</th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">x</th>
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    <tr><td class="gt_row gt_left">1</td>
<td class="gt_row gt_left">0.42</td></tr>
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<td class="gt_row gt_left">0.23</td></tr>
    <tr><td class="gt_row gt_left">10</td>
<td class="gt_row gt_left">0.18</td></tr>
    <tr><td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">20</td>
<td class="gt_row gt_left" style="border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: transparent;">?</td></tr>
  </tbody>
  
  
</table>
</div>

<ol type = "a">
  <li>0.35</li>
  <li>0.17</li>
  <li>0.40</li>
  <li>Cannot be determined given the information</li>
</ol>

---
class: clear,middle

### Continuous

---
# Defining Probability Density Function
Let `$Y$` be a continuous random variable defined over the interval `$[a,b]$`.

The .hi.purple[probability density function] (or pdf) of `$Y$` is a function, `$f_Y(y)$`, that assigns a probability value to every possible .it[interval] in `$[a,b]$`.

We write
$$ Pr(c\leqslant Y\leqslant d) = \int_{c}^{d}f_Y(y) dy, $$
for all `$(c,d)\subset [a,b]$`.

---
# Example pdf

For `$Z \sim N(0,1)$`, find `$P(Z <= -1)$`.

---
# Integral of PDF = Probability

---
# Example PDF
Suppose that `$Y$` is a continuous random variable with pdf `$f_Y(y) = 3y^2$` for `$0<y<1$`. What is `$P(\frac{1}{4} \leqslant Y \leqslant \frac{1}{2})$`?

---
# Example PDF
Suppose that `$Y$` is a continuous random variable with pdf `$f_Y(y) = 3y^2$` for `$0<y<1$`. What is `$P(\frac{1}{4} \leqslant Y \leqslant \frac{1}{2})$`?

---
# Properties of PDFs
We say a `$f_Y(y)$` is a .hi.purple[valid] pdf if it satisfies the following:

1. `$0\leqslant \int f_Y(y) \leqslant 1$` for all `$y \in [a,b]$`.

2. `$\displaystyle\int_{a}^b f_Y(y)dy = 1.$`

Note that `$Pr(Y=a) = \displaystyle\int_{a}^a f_Y(y)dy = 0$`.

At first this might seem counterintuitive. But imagine trying to stop a stopwatch at exactly 30 seconds. What is the probability of that event?

---
# Clicker Question
Given the pdf, `$f(y)=3y^2$` for `$0<y<1$`. What is the `$P(Y<1/3)$`?

---
# Midterm Example
Consider the probability distribution for random variable Y:

$$ 
f(y)= 8y, \ 0 \leq y \leq \frac{1}{2} 
$$

1. Find `$P(Y<\frac{1}{3})$`
2. Find `$P(Y=\frac{1}{4})$`
3. Find `$P(\frac{3}{4}<Y<1)$`