Axiom:Axioms of Uncertainty/Axiom 2

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Let $Z$ be a random variable.

Let $Z$ take the values $a_i$ with probability $p_i$, where $i \in \set {1, 2, \ldots, n}$.

Let $H_n: Z^n \to \R$ be a mapping which is to be defined as the uncertainty of $Z$.

$H_n$ fulfils the following axiom:

For any permutation $\pi$ of $\tuple {1, 2, \dotsc, n}$:

$\map {H_n} {p_1, p_2, \ldots, p_n} = \map {H_n} {p_{\map \pi 1}, p_{\map \pi 2}, \ldots, p_{\map \pi n} }$

That is, $H_n$ is a symmetric function of the arguments $p_1, p_2, \dotsc, p_n$.