Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability ▷ Exceedance probability

Definition: Let $X = \left\lbrace X_1, \ldots, X_n \right\rbrace$ be a set of $n$ random variables which the joint probability distribution $p(X) = p(X_1, \ldots, X_n)$. Then, the exceedance probability for random variable $X_i$ is the probability that $X_i$ is larger than all other random variables $X_j, \; j \neq i$:

\[\label{eq:EP} \begin{split} \varphi(X_i) &= \mathrm{Pr}\left( \forall j \in \left\lbrace 1, \ldots, n | j \neq i \right\rbrace: \, X_i > X_j \right) \\ &= \mathrm{Pr}\left( \bigwedge_{j \neq i} X_i > X_j \right) \\ &= \mathrm{Pr}\left( X_i = \mathrm{max}(\left\lbrace X_1, \ldots, X_n \right\rbrace) \right) \\ &= \int_{X_i = \mathrm{max}(X)} p(X) \, \mathrm{d}X \; . \end{split}\]
 
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Metadata: ID: D103 | shortcut: prob-exc | author: JoramSoch | date: 2020-10-22, 04:36.