More secratary edits

Advanced/Stopping Problems/parts/1 secretary.tex

@@ -79,7 +79,7 @@ We have no absolute metric by which to judge each candidate.
 
 \vspace{2mm}
 
-Thus, all $I_x$ defined above are independent: \par
+Thus, all $I_x$ defined above are independent:
 the outcome of any $I_a$ does not influence the probabilities of any other $I_b$.
 
 \vspace{2mm}
@@ -92,4 +92,48 @@ the results of past $I_x$ cannot possibly provide information about future $I_x$
 
 Given the above realizations, we are left with only one kind of strategy: \par
 We blindly reject the first $k$ applicants, and select the first \say{best-seen} applicant we encounter afterwards.
-All we need to do now is pick the optimal $k$.
+All we need to do now is pick the optimal $k$.
+
+\pagebreak
+
+\problem{}
+Consider the secretary problem with a given $n$. \par
+What is the probability distribution of each of $I_1, I_2, ..., I_n$? \par
+\note{That is, what are $\mathcal{P}(I_x = \texttt{0})$ and $\mathcal{P}(I_x = \texttt{1})$ for each $x$?}
+
+\vfill
+
+\problem{}
+What is the probability that the $x^\text{th}$ applicant is \textit{not} the best-seen applicant? \par
+In other words, what is the probability we reject the $x^\text{th}$ candidate? \par
+\note{Assuming that $x > k$. Otherwise, the probability of rejection is 1!}
+
+\vfill
+
+\problem{}<phisubn>
+Again, consider the secretary problem with a fixed $n$. \par
+If we reject the first $k$ applicants and hire the first \say{best-yet} applicant we encounter, \par
+what is the probability that we select the best candidate? \par
+Call this function $\phi_n(k)$.
+
+\vfill
+
+\problem{}
+Find $
+	\underset{x \in \{0, ..., n\}}{\text{max}}
+	\Bigl(\phi_n(k)\Bigr)
+$ for all $n$ in $\{1, 2, 3, 4, 5\}$.
+
+\vfill
+
+\problem{}
+Find $
+	\underset{x \in \mathbb{R}}{\text{max}}
+	\Bigl(~
+		\underset{n \rightarrow \infty}{\text{lim}}
+		\phi_n(k)
+	~\Bigr)
+$
+
+
+\vfill