@ -46,17 +46,17 @@ a \textit{random variable} is a function from $\Omega$ to a specified output set
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For example, given the three-coin-toss sample space
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$\Omega = \{
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\texttt{TTT},~ \texttt{TTH},~ \texttt{THT},~
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\texttt{THH},~ \texttt{HTT},~ \texttt{HTH},~
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\texttt{HHT},~ \texttt{HHH}
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\texttt{TTT},~ \texttt{TTH},~ \texttt{THT},~ % spell:disable-line
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\texttt{THH},~ \texttt{HTT},~ \texttt{HTH},~ % spell:disable-line
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\texttt{HHT},~ \texttt{HHH} % spell:disable-line
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\}$,
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We can define a random variable $\mathcal{H}$ as \say{the number of heads in a throw of three coins}. \par
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As a function, $\mathcal{H}$ maps values in $\Omega$ to values in $\mathbb{Z}^+_0$ and is defined as:
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\begin{itemize}
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\item $\mathcal{H}(\texttt{TTT}) = 0$
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\item $\mathcal{H}(\texttt{TTH}) = 1$
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\item $\mathcal{H}(\texttt{THT}) = 1$
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\item $\mathcal{H}(\texttt{THH}) = 2$
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\item $\mathcal{H}(\texttt{TTT}) = 0$ % spell:disable-line
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\item $\mathcal{H}(\texttt{TTH}) = 1$ % spell:disable-line
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\item $\mathcal{H}(\texttt{THT}) = 1$ % spell:disable-line
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||||
\item $\mathcal{H}(\texttt{THH}) = 2$ % spell:disable-line
|
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\item ...and so on.
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\end{itemize}
|
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|
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@ -70,7 +70,7 @@ the set of outcomes that produce that value. \par
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||||
\vspace{2mm}
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For example, if we wanted to compute $\mathcal{P}(\mathcal{H} = 2)$, we would find
|
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$\mathcal{P}\bigl(\{\texttt{THH}, \texttt{HTH}, \texttt{HHT}\}\bigr)$.
|
||||
$\mathcal{P}\bigl(\{\texttt{THH}, \texttt{HTH}, \texttt{HHT}\}\bigr)$. % spell:disable-line
|
||||
|
||||
|
||||
\problem{}
|
||||
|
@ -191,7 +191,7 @@ what is the probability that we select the best candidate? \par
|
||||
Call this probability $\phi_n(k)$.
|
||||
|
||||
\begin{solution}
|
||||
Using \ref{seca} and \ref{secb}, this is straightfoward:
|
||||
Using \ref{seca} and \ref{secb}, this is straightforward:
|
||||
\[
|
||||
\phi_n(k)
|
||||
= \sum_{x = k}^{n}\left( \frac{1}{n} \times \frac{k-1}{x-1} \right)
|
||||
|
Reference in New Issue
Block a user