diff --git a/Advanced/Stopping Problems/main.tex b/Advanced/Stopping Problems/main.tex index 7ed267d..7b32d45 100755 --- a/Advanced/Stopping Problems/main.tex +++ b/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} diff --git a/Advanced/Stopping Problems/parts/0 review.tex b/Advanced/Stopping Problems/parts/0 probability.tex similarity index 94% rename from Advanced/Stopping Problems/parts/0 review.tex rename to Advanced/Stopping Problems/parts/0 probability.tex index 0cf2724..2ba3aec 100644 --- a/Advanced/Stopping Problems/parts/0 review.tex +++ b/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{} 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{} diff --git a/Advanced/Stopping Problems/parts/1 intro.tex b/Advanced/Stopping Problems/parts/1 intro.tex index 98b9d24..90b0250 100644 --- a/Advanced/Stopping Problems/parts/1 intro.tex +++ b/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{} 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} diff --git a/Advanced/Stopping Problems/parts/2 secretary.tex b/Advanced/Stopping Problems/parts/2 secretary.tex index c402c94..ac9b823 100644 --- a/Advanced/Stopping Problems/parts/2 secretary.tex +++ b/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.} diff --git a/Advanced/Stopping Problems/parts/3 orderstat.tex b/Advanced/Stopping Problems/parts/3 orderstat.tex index 9d0dca3..017ec1f 100644 --- a/Advanced/Stopping Problems/parts/3 orderstat.tex +++ b/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{} -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