Tom edits

Advanced/Stopping Problems/main.tex

@@ -18,7 +18,7 @@
 
 	\maketitle
 
-	\input{parts/0 review.tex}
+	\input{parts/0 probability.tex}
 	\input{parts/1 intro.tex}
 	\input{parts/2 secretary.tex}
 	\input{parts/3 orderstat.tex}

Advanced/Stopping Problems/parts/0 probability.tex

@@ -1,4 +1,4 @@
-\section{Review}
+\section{Probability}
 
 \definition{}
 A \textit{sample space} is a finite set $\Omega$. \par
@@ -16,15 +16,14 @@ Any probability function has the following properties:
 \end{itemize}
 
 
-\problem{}
+\problem{}<threecoins>
 Say we flip a fair coin three times. \par
 List all elements of the sample space $\Omega$ this experiment generates.
 
 \vfill
 
 \problem{}
-Again, flip a fair coin three times. \par
-Find the following:
+Using the same setup as \ref{threecoins}, find the following:
 \begin{itemize}
 	\item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has at least two \say{heads}}\} ~)$
 	\item $\mathcal{P}(~ \{\omega \in \Omega ~|~ \omega \text{ has an odd number of \say{heads}}\} ~)$
@@ -94,7 +93,7 @@ Find $\mathcal{P}(\mathcal{X} = x)$ for all $x$ in $\mathbb{Z}$.
 
 
 \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
 \begin{equation*}
 	\mathcal{E}(\mathcal{X})
@@ -118,7 +117,7 @@ Show that $\mathcal{E}(\mathcal{A} + \mathcal{B}) = \mathcal{E}(\mathcal{A}) + \
 
 \definition{}
 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.
 
 \definition{}

Advanced/Stopping Problems/parts/1 intro.tex

@@ -3,7 +3,7 @@
 \generic{Setup:}
 Suppose we toss a 6-sided die $n$ times. \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>
 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
 
 \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{tabular}{|| c | c | c ||}
@@ -53,7 +54,8 @@ For what value of $l$ is the probability in \ref{lastl} maximal?
 
 \problem{}
 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?
 
 \begin{solution}

Advanced/Stopping Problems/parts/2 secretary.tex

@@ -229,20 +229,18 @@ Let $r = \frac{k-1}{n}$, the fraction of applicants we reject. Show that
 \vfill
 
 \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*}
-	\underset{n \rightarrow \infty}{\text{lim}}
 	\sum_{x=k}^{n}\frac{1}{x-1}
 	~\approx~ \text{ln}\Bigl(\frac{n}{k}\Bigr)
 \end{equation*}
-Use this fact to find $\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k)$.~
-\hint{For large $n$, $\frac{k-1}{n} \approx \frac{k}{n}$.}
+Use this fact to find an approximation of $\phi_n(k)$ at large $n$ in terms of $r$. \par
+\hint{If $n$ is big, $\frac{k-1}{n} \approx \frac{k}{n}$.}
 
 \begin{solution}
 	\begin{equation*}
-		\underset{n \rightarrow \infty}{\text{lim}} \phi_n(k)
-		~=~ \underset{n \rightarrow \infty}{\text{lim}}
-			\Biggl( r \sum_{x = k}^{n}\left( \frac{1}{x-1} \right) \Biggr)
+		\phi_n(k)
+		~=~  r \sum_{x = k}^{n}\left( \frac{1}{x-1} \right)
 		~\approx~ r \times \text{ln}\left(\frac{n}{k}\right)
 		~=~ -r \times \text{ln}\left(\frac{k}{n}\right)
 		~\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
 
 \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
 \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
 
-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
 \note{Recall that the random strategy succeeds with probability $\nicefrac{1}{n}$. \par
 That is, it quickly becomes small as $n$ gets large.}

Advanced/Stopping Problems/parts/3 orderstat.tex

@@ -85,14 +85,13 @@ Given some $y$, what is the probability that all five $\mathcal{X}_i$ are smalle
 
 
 \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
 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>
-Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$. \par
-We take $5$ observations of $\mathcal{X}$. \par
+Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$, of which we take $5$ observations. \par
 Given some $y$, what is the probability that $x_5 < y$? How about $x_4 <y $?
 
 \begin{solution}