Let's consider three random variables $\xi_1, \xi_2, \xi_3$ distributed over $\{1, 2, 3, \ldots, M\}$ values. Is it possible that $P(\xi_1 < \xi_3) > 0.5, P(\xi_3 < \xi_2) > 0.5, P(\xi_2 < \xi_1) > 0.5$? So we cannot order them using probabilities. Thank you.
ordering three random variables
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probability
1 Answers
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Let $\Omega := [0,1]$ and $\mathbb{P} := \lambda|_{[0,1]}$ the lebesgue measure on $[0,1]$. Define
$\xi_1(x) := \begin{cases}3 & x \in \left[0,\frac{1}{3} \right] \\ 1 & x \in \left(\frac{1}{3},\frac{2}{3} \right) \\ 2 & x \in \left[\frac{2}{3},1 \right] \end{cases} \\ \xi_2(x) := \begin{cases}2 & x \in \left[0,\frac{1}{3} \right] \\ 3& x \in \left(\frac{1}{3},\frac{2}{3} \right) \\ 1 & x \in \left[\frac{2}{3},1 \right] \end{cases} \\ \xi_3(x) := \begin{cases}1 & x \in \left[0,\frac{1}{3} \right] \\ 2 & x \in \left(\frac{1}{3},\frac{2}{3} \right) \\ 3 & x \in \left[\frac{2}{3},1 \right] \end{cases}$
Then $\mathbb{P}[\xi_1 < \xi_3] = \mathbb{P}[\xi_3<\xi_2] = \mathbb{P}[\xi_2<\xi_1] = \frac{2}{3}$.
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0I see it why this is a case – 2012-12-06