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Mark 2024-09-26 09:25:14 -07:00
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Advanced/Stopping Problems/parts/3 orderstat.tex
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@ -88,7 +88,7 @@ Given some $y$, what is the probability that all five $\mathcal{X}_i$ are smalle
Say we have a random variable $\mathcal{X}$ which we observe $n$ times. \note{(for example, we repeatedly roll a die)} 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 $\mathcal{X}$.
a
\problem{}<ostatone> \problem{}<ostatone>
Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$, of which we take $5$ observations. \par Say we have a random variable $\mathcal{X}$ uniformly distributed on $[0, 1]$, of which we take $5$ observations. \par
@ -188,10 +188,6 @@ reject the first $e^{-1} \times n$ candidates, and select the next \say{best-yet
How effective is this strategy for the ranked secretary problem? \par How effective is this strategy for the ranked secretary problem? \par
Find the expected rank of the applicant we select using this strategy. Find the expected rank of the applicant we select using this strategy.
\begin{solution}
Coming soon.
\end{solution}
\vfill \vfill
@ -200,7 +196,8 @@ Assuming we use the same kind of strategy as before (reject $k$, select the next
show that $k = \sqrt{n}$ optimizes the expected rank of the candidate we select. show that $k = \sqrt{n}$ optimizes the expected rank of the candidate we select.
\begin{solution} \begin{solution}
This is a difficult bonus problem. Solution coming later. This is a difficult bonus problem. see
\texttt{Neil Bearden, J. (2006). A new secretary problem with rank-based selection and cardinal payoffs.}
\end{solution} \end{solution}
\vfill \vfill