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Mark 2024-09-25 09:13:56 -07:00
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2
Advanced/Stopping Problems/main.tex
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@ -18,7 +18,7 @@
\maketitle \maketitle
\input{parts/0 review.tex} \input{parts/0 probability.tex}
\input{parts/1 intro.tex} \input{parts/1 intro.tex}
\input{parts/2 secretary.tex} \input{parts/2 secretary.tex}
\input{parts/3 orderstat.tex} \input{parts/3 orderstat.tex}

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Advanced/Stopping Problems/parts/0 review.tex → Advanced/Stopping Problems/parts/0 probability.tex
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@ -1,4 +1,4 @@
\section{Review} \section{Probability}
\definition{} \definition{}
A \textit{sample space} is a finite set $\Omega$. \par A \textit{sample space} is a finite set $\Omega$. \par
@ -16,15 +16,14 @@ Any probability function has the following properties:
\end{itemize} \end{itemize}
\problem{} \problem{}<threecoins>
Say we flip a fair coin three times. \par Say we flip a fair coin three times. \par
List all elements of the sample space $\Omega$ this experiment generates. List all elements of the sample space $\Omega$ this experiment generates.
\vfill \vfill
\problem{} \problem{}
Again, flip a fair coin three times. \par Using the same setup as \ref{threecoins}, find the following:
Find the following:
\begin{itemize} \begin{itemize}
\item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has at least two \say{heads}}\} ~)$ \item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has at least two \say{heads}}\} ~)$
\item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has an odd number of \say{heads}}\} ~)$ \item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has an odd number of \say{heads}}\} ~)$
@ -94,7 +93,7 @@ Find $\mathcal{P}(\mathcal{X} = x)$ for all $x$ in $\mathbb{Z}$.
\definition{} \definition{}
Say we have a random variable that produces outputs in a set $A$. \par Say we have a random variable $\mathcal{X}$ that produces outputs in $\mathbb{R}$. \par
The \textit{expected value} of $\mathcal{X}$ is then defined as The \textit{expected value} of $\mathcal{X}$ is then defined as
\begin{equation*} \begin{equation*}
\mathcal{E}(\mathcal{X}) \mathcal{E}(\mathcal{X})
@ -118,7 +117,7 @@ Show that $\mathcal{E}(\mathcal{A} + \mathcal{B}) = \mathcal{E}(\mathcal{A}) + \
\definition{} \definition{}
Let $A$ and $B$ be events on a sample space $\Omega$. \par Let $A$ and $B$ be events on a sample space $\Omega$. \par
We say that $A$ and $B$ are \textit{independent} if $\mathcal{E}(A \cap B) = \mathcal{P}(A) + \mathcal{P}(B)$. \par We say that $A$ and $B$ are \textit{independent} if $\mathcal{P}(A \cap B) = \mathcal{P}(A) + \mathcal{P}(B)$. \par
Intuitively, events $A$ and $B$ are independent if the outcome of one does not affect the other. Intuitively, events $A$ and $B$ are independent if the outcome of one does not affect the other.
\definition{} \definition{}

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Advanced/Stopping Problems/parts/1 intro.tex
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@ -3,7 +3,7 @@
\generic{Setup:} \generic{Setup:}
Suppose we toss a 6-sided die $n$ times. \par Suppose we toss a 6-sided die $n$ times. \par
It is easy to detect the first time we roll a 6. \par It is easy to detect the first time we roll a 6. \par
What should we do if we want to annouce the \textit{last}? What should we do if we want to detect the \textit{last}?
\problem{}<lastl> \problem{}<lastl>
Given $l \leq n$, what is the probability that the last $l$ Given $l \leq n$, what is the probability that the last $l$
@ -17,7 +17,8 @@ tosses of this die contain exactly one six? \par
\vfill \vfill
\problem{} \problem{}
For what value of $l$ is the probability in \ref{lastl} maximal? For what value of $l$ is the probability in \ref{lastl} maximal? \par
The following table may help.
\begin{center} \begin{center}
\begin{tabular}{|| c | c | c ||} \begin{tabular}{|| c | c | c ||}
@ -53,7 +54,8 @@ For what value of $l$ is the probability in \ref{lastl} maximal?
\problem{} \problem{}
Finish your solution: \par Finish your solution: \par
In $n$ rolls of a six-sided die, when should we announce the last time we roll a 6? \par In $n$ rolls of a six-sided die, what strategy maximizes
our chance of detecting the last $6$ that is rolled? \par
What is the probability of our guess being right? What is the probability of our guess being right?
\begin{solution} \begin{solution}

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Advanced/Stopping Problems/parts/2 secretary.tex
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@ -229,20 +229,18 @@ Let $r = \frac{k-1}{n}$, the fraction of applicants we reject. Show that
\vfill \vfill
\problem{} \problem{}
With a bit of faily unpleasant calculus, we can show that With a bit of faily unpleasant calculus, we can show that the following is true for large $n$:
\begin{equation*} \begin{equation*}
\underset{n \rightarrow \infty}{\text{lim}}
\sum_{x=k}^{n}\frac{1}{x-1} \sum_{x=k}^{n}\frac{1}{x-1}
~\approx~ \text{ln}\Bigl(\frac{n}{k}\Bigr) ~\approx~ \text{ln}\Bigl(\frac{n}{k}\Bigr)
\end{equation*} \end{equation*}
Use this fact to find $\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k)$.~ Use this fact to find an approximation of $\phi_n(k)$ at large $n$ in terms of $r$. \par
\hint{For large $n$, $\frac{k-1}{n} \approx \frac{k}{n}$.} \hint{If $n$ is big, $\frac{k-1}{n} \approx \frac{k}{n}$.}
\begin{solution} \begin{solution}
\begin{equation*} \begin{equation*}
\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k) \phi_n(k)
~=~ \underset{n \rightarrow \infty}{\text{lim}} ~=~ r \sum_{x = k}^{n}\left( \frac{1}{x-1} \right)
\Biggl( r \sum_{x = k}^{n}\left( \frac{1}{x-1} \right) \Biggr)
~\approx~ r \times \text{ln}\left(\frac{n}{k}\right) ~\approx~ r \times \text{ln}\left(\frac{n}{k}\right)
~=~ -r \times \text{ln}\left(\frac{k}{n}\right) ~=~ -r \times \text{ln}\left(\frac{k}{n}\right)
~\approx~ -r \times \text{ln}(r) ~\approx~ -r \times \text{ln}(r)
@ -252,7 +250,7 @@ Use this fact to find $\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k)$.~
\vfill \vfill
\problem{} \problem{}
Find the $k$ that maximizes $\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k)$. \par Find the $r$ that maximizes $\underset{n \rightarrow \infty}{\text{lim}} \phi_n$. \par
Also, find the value of $\phi_n$ at this point. \par Also, find the value of $\phi_n$ at this point. \par
\note{If you aren't familiar with calculus, ask an instructor for help.} \note{If you aren't familiar with calculus, ask an instructor for help.}
@ -270,7 +268,7 @@ Also, find the value of $\phi_n$ at this point. \par
\vfill \vfill
Following this strategy, we should thus expect to select the best candidate about $e^{-1} = 37\%$ of the time, Thus, the \say{look-then-leap} strategy with $r = e^{-1}$ should select the best candidate about $e^{-1} = 37\%$ of the time,
\textit{regardless of $n$.} Our probability of success does not change as $n$ gets larger! \par \textit{regardless of $n$.} Our probability of success does not change as $n$ gets larger! \par
\note{Recall that the random strategy succeeds with probability $\nicefrac{1}{n}$. \par \note{Recall that the random strategy succeeds with probability $\nicefrac{1}{n}$. \par
That is, it quickly becomes small as $n$ gets large.} That is, it quickly becomes small as $n$ gets large.}

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Advanced/Stopping Problems/parts/3 orderstat.tex
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@ -85,14 +85,13 @@ Given some $y$, what is the probability that all five $\mathcal{X}_i$ are smalle
\definition{} \definition{}
Say we have a random variable $\mathcal{X}$ which we observe $n$ times. \par Say we have a random variable $\mathcal{X}$ which we observe $n$ times. \note{(for example, we repeatedly roll a die)}
We'll arrange these observations in increasing order, labeled $x_1 < x_2 < ... < x_n$. \par We'll arrange these observations in increasing order, labeled $x_1 < x_2 < ... < x_n$. \par
Under this definition, $x_i$ is called the \textit{$i^\text{th}$ order statistic}---the $i^\text{th}$ smallest sample of $\mathcal{X}$. Under this definition, $x_i$ is called the \textit{$i^\text{th}$ order statistic}---the $i^\text{th}$ smallest sample of $\mathcal{X}$.
\problem{}<ostatone> \problem{}<ostatone>
Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$. \par Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$, of which we take $5$ observations. \par
We take $5$ observations of $\mathcal{X}$. \par
Given some $y$, what is the probability that $x_5 < y$? How about $x_4 <y $? Given some $y$, what is the probability that $x_5 < y$? How about $x_4 <y $?
\begin{solution} \begin{solution}