A question related to cardinality and probability(repost).
Dear all,
I have a question below related to both probability and cardinality. Let me know if my formulation of the problem is non-rigorous or confusing. Any proof or suggestions are appreciated. Thank you all.
The question follows.
Consider a set \(\displaystyle I\) consists of \(\displaystyle N\) incidents.
\[I=\{i_{1},i_{2},...,i_{k},...i_{N}\}\]
Each incident has a probability to happen, i.e. incident \(\displaystyle i_{k}\) happens with the probability \(\displaystyle r_{k}\). Without loss of generality, we assume \(\displaystyle r_{1}\geq r_{2}\geq ... \geq r_{k}\geq ... \geq r_{N}\)
Given a constant \(\displaystyle n<N\), we can have set \(\displaystyle I_{1}=\{i_{1},i_{2},...,i_{n}\}\). Apparently, \(\displaystyle |I_{1}|=n\) and \(\displaystyle I_{1}\subset I\).
Define a mapping \(\displaystyle I\to S\) with \(\displaystyle S=\{s_{1},s_{2},...,s_{k},...s_{N}\}\) subject to
\[
s_{k} = \left\{ \begin{array}{ccc}
1 &\mbox{ (Pr=$r_{k}$)} \\
0 &\mbox{ (Pr=$1-r_{k}$)} \\
\end{array} \right.
\]
Pick out the incidents with correspond \(\displaystyle s\) being 1 to form the set \(\displaystyle I_{2}\) , i.e.
\[I_{2}=\{i_{m_{1}},i_{m_{2}},...,i_{m_{M}}\} \quad \mbox{and} \quad s_{m_{k}}=1 \quad k=1,2,...,M \]
Apparently, \(\displaystyle |I_{2}|=M\) and \(\displaystyle I_{2}\subset I\). Note that there could be \(\displaystyle I_{2}\ne I_{1}\) and \(\displaystyle |I_{2}| \ne |I_{1}|\).
The question is,
If we have two set \(\displaystyle A\) and \(\displaystyle B\) with \(\displaystyle |A|=|B|=n\) and assume
\[ |A \cap I_{1}| \geq |B \cap I_{1}| \]
Is the following statement true?
\[ E(|A \cap I_{2}|) \geq E(|B \cap I_{2}|) \]
where \(\displaystyle E\) means expected value.
If this is true, how to prove it? If not, how to prove it’s not true?
Dear all,
I have a question below related to both probability and cardinality. Let me know if my formulation of the problem is non-rigorous or confusing. Any proof or suggestions are appreciated. Thank you all.
The question follows.
Consider a set \(\displaystyle I\) consists of \(\displaystyle N\) incidents.
\[I=\{i_{1},i_{2},...,i_{k},...i_{N}\}\]
Each incident has a probability to happen, i.e. incident \(\displaystyle i_{k}\) happens with the probability \(\displaystyle r_{k}\). Without loss of generality, we assume \(\displaystyle r_{1}\geq r_{2}\geq ... \geq r_{k}\geq ... \geq r_{N}\)
Given a constant \(\displaystyle n<N\), we can have set \(\displaystyle I_{1}=\{i_{1},i_{2},...,i_{n}\}\). Apparently, \(\displaystyle |I_{1}|=n\) and \(\displaystyle I_{1}\subset I\).
Define a mapping \(\displaystyle I\to S\) with \(\displaystyle S=\{s_{1},s_{2},...,s_{k},...s_{N}\}\) subject to
\[
s_{k} = \left\{ \begin{array}{ccc}
1 &\mbox{ (Pr=$r_{k}$)} \\
0 &\mbox{ (Pr=$1-r_{k}$)} \\
\end{array} \right.
\]
Pick out the incidents with correspond \(\displaystyle s\) being 1 to form the set \(\displaystyle I_{2}\) , i.e.
\[I_{2}=\{i_{m_{1}},i_{m_{2}},...,i_{m_{M}}\} \quad \mbox{and} \quad s_{m_{k}}=1 \quad k=1,2,...,M \]
Apparently, \(\displaystyle |I_{2}|=M\) and \(\displaystyle I_{2}\subset I\). Note that there could be \(\displaystyle I_{2}\ne I_{1}\) and \(\displaystyle |I_{2}| \ne |I_{1}|\).
The question is,
If we have two set \(\displaystyle A\) and \(\displaystyle B\) with \(\displaystyle |A|=|B|=n\) and assume
\[ |A \cap I_{1}| \geq |B \cap I_{1}| \]
Is the following statement true?
\[ E(|A \cap I_{2}|) \geq E(|B \cap I_{2}|) \]
where \(\displaystyle E\) means expected value.
If this is true, how to prove it? If not, how to prove it’s not true?
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