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andy
2026-01-13 07:06:29 -05:00
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publication/publication.ptx
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<!-- Whether to knowl a particular elements is set here --> <!-- Whether to knowl a particular elements is set here -->
<!-- Lots of elements have this possibility; see the guide --> <!-- Lots of elements have this possibility; see the guide -->
<!--<knowl <knowl
theorem="no" theorem="no"
proof="yes" proof="yes"
definition="no" definition="no"
example="yes" example="no"
example-solution="yes" example-solution="yes"
project="no" project="no"
task="no" task="no"
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exercise-divisional="no" exercise-divisional="no"
exercise-worksheet="no" exercise-worksheet="no"
exercise-readingquestion="no" exercise-readingquestion="no"
/>--> />
<!-- Specify the theme for the html by giving names to --> <!-- Specify the theme for the html by giving names to -->
<!-- override defaults. Ex:<css theme="denver"/> --> <!-- override defaults. Ex:<css theme="denver"/> -->
source/main.ptx
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<!-- Other documents could go here, like a schedule, project description, etc. --> <!-- Other documents could go here, like a schedule, project description, etc. -->
</chapter> </chapter>
<chapter xml:id="ch-notes">
<title>Class Notes</title>
<xi:include href="./notes/week01.ptx" />
</chapter>
<chapter xml:id="recitations"> <chapter xml:id="recitations">
<title>Recitations</title> <title>Recitations</title>
source/notes/week01.ptx
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<introduction> <introduction>
<p> <p>
This is an outline of the topics we covered in the first week of class. This is an outline of the topics we covered in the first week of class.
These notes are <em>not</em> a substitute for your own note-taking.
I highly recommend that you take your own notes during class.
If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
</p> </p>
</introduction> </introduction>
<!-- Can also be <handout> instead of <subsection> to get a printable version -->
<subsection> <subsection>
<title>Monday 8/22</title> <title>Tuesday 1/13</title>
<p>
In calculus, you mostly asked <term>deterministic</term> questions about functions.
In science experiments, you need to take <term>randomness</term> into account.
</p>
<example>
<statement>
<p>
A toxin molecule in a cell has a certain chance each minute to leave the cell.
</p>
</statement>
</example>
<example>
<statement>
<p>
A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result.
</p>
</statement>
</example>
<example>
<statement>
<p>
Based on historical year-over-year population growth rate data for a particular species, what is a reasonable range for next year's population?
</p>
</statement>
</example>
<definition xml:id="def-sample-space">
<statement>
<p>
The <term>sample space</term>, often denoted <m>\Omega</m>, is the set of all possible results of an experiment.
A single result is called an <term>outcome</term>, while a collection of results is called an <term>event</term>.
</p>
</statement>
</definition>
<definition xml:id="def-subset">
<statement>
<p>
Let <m>A</m> be a set.
The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>.
Given another set <m>B</m>, we say <m>A</m> is a <term>subset</term> of <m>B</m>, written <m>A\subset B</m>, to mean that every element of the set <m>A</m> is also an element of the set <m>B</m>.
</p>
</statement>
</definition>
<example xml:id="example-sample-space">
<statement>
<p>
An experiment consists of rolling a standard 6-sided die.
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even.
</p>
</statement>
</example>
<definition xml:id="def-set-operations">
<statement>
<p>
Consider sets <m>A</m> and <m>B</m>, each contained inside <m>\Omega</m>.
We can combine sets in a variety of ways: <dl>
<li>
<title>Union</title>
<p>
The <term>union</term> of <m>A</m> and <m>B</m> is the set <m>A \cup B = \{x \mid x \in A \text{ or } x \in B\}</m>.
</p>
</li>
<li>
<title>Intersection</title>
<p>
The <term>intersection</term> of <m>A</m> and <m>B</m> is the set <m>A \cap B = \{x \mid x \in A \text{ and } x \in B\}</m>.
</p>
</li>
<li>
<title>Difference</title>
<p>
The <term>set difference</term> <m>A-B</m> is the set <m>A - B = \{x \mid x \in A \text{ and } x \notin B\}</m>.
</p>
</li>
<li>
<title>Complement</title>
<p>
The <term>complement</term> of <m>A</m> is the set <m>A^c = \{x \in \Omega \mid x \notin A\}</m>.
</p>
</li>
<li>
<title>Empty Set</title>
<p>
The <term>empty set</term>, usually written <m>\emptyset</m> or <m>\{\}</m>, is the set which contains no elements.
</p>
</li>
</dl>
</p>
</statement>
</definition>
<p>
It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing sets:
</p>
<figure xml:id="fig-Venn-diagram">
<caption>Example Venn Diagram</caption>
<image width="50%">
<description>
<p>
Venn diagram showing sets <m>A, B, C</m> with the region representing <m>(A\cup B\cup C) - (A \cap C)</m> shaded.
</p>
</description>
<latex-image>
\begin{tikzpicture}
\def\firstcircle{(90:1.75cm) circle (2.5cm)}
\def\secondcircle{(210:1.75cm) circle (2.5cm)}
\def\thirdcircle{(330:1.75cm) circle (2.5cm)}
\fill[gray!30] \firstcircle;
\fill[gray!30] \secondcircle;
\fill[gray!30] \thirdcircle;
\begin{scope}
\clip \firstcircle;
\clip \thirdcircle;
\fill[white] \firstcircle;
\end{scope}
\draw \firstcircle node[text=black,above] {$A$};
\draw \secondcircle node [text=black,below left] {$B$};
\draw \thirdcircle node [text=black,below right] {$C$};
\node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$};
\end{tikzpicture}
</latex-image>
</image>
</figure>
<p>
Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.
</p>
<example>
<statement>
<p>
An experiment consists of rolling a standard 6-sided die.
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
We might assign probabilities as follows:
</p>
<table>
<title>Distribution for a fair die</title>
<tabular halign="center">
<row bottom="minor">
<cell><m>x</m></cell>
<cell><m>\Pr(x)</m></cell>
</row>
<row>
<cell>1</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>2</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>3</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>4</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>5</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>6</cell>
<cell><m>1/6</m></cell>
</row>
</tabular>
</table>
<p>
Note that we don't have to assign the same probability to each outcome.
If we do, we call this distribution <term>uniform</term>.
If we have some event, such as <m>A = \{2, 4, 6\}</m>, then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event:
<md>
<mrow> \Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} </mrow>
</md>
Note that this is the same result that we would get if we counted the total number of outcomes in <m>A</m> and divided by the total number of outcomes in <m>\Omega</m>.
</p>
</statement>
</example>
<definition xml:id="def-cardinality">
<statement>
<p>
Given a finite set <m>A</m>, the <term>cardinality</term> of <m>A</m>, written <m>|A|</m>, is the number of elements in <m>A</m>.
</p>
</statement>
</definition>
<fact>
<statement>
<p>
If <m>\Omega</m> is a finite probability space with the uniform distribution and <m>A \subset \Omega</m> is an event, then:
<md>
<mrow> \Pr(A) = \frac{|A|}{|\Omega|} </mrow>
</md>
</p>
</statement>
</fact>
<example>
<statement>
<p>
Suppose we have a weighted die that's much more likely to come up 6 than any other outcome.
</p>
<table>
<title>Distribution for a fair die</title>
<tabular halign="center">
<row bottom="minor">
<cell><m>x</m></cell>
<cell><m>\Pr(x)</m></cell>
</row>
<row>
<cell>1</cell>
<cell>0.1</cell>
</row>
<row>
<cell>2</cell>
<cell>0.1</cell>
</row>
<row>
<cell>3</cell>
<cell>0.1</cell>
</row>
<row>
<cell>4</cell>
<cell>0.2</cell>
</row>
<row>
<cell>5</cell>
<cell>0.1</cell>
</row>
<row>
<cell>6</cell>
<cell>0.4</cell>
</row>
</tabular>
</table>
<p>
With this distribution, if <m>A = \{2, 4, 6\}</m>, then:
<md>
<mrow> \Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}. </mrow>
</md>
</p>
</statement>
</example>
</subsection>
<!--
<subsection>
<title>Thursday 1/15</title>
<p> <p>
</p> </p>
</subsection> </subsection>
-->
</section>
<subsection>
<title>Wednesday 8/24</title>
<p>
</p>
</subsection>
<subsection>
<title>Friday 8/26</title>
<p>
</p>
</subsection>
</section>