Axiom:Axioms of Uncertainty/Axiom 5
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Axiom
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:
- $\map {H_n} {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n} \le \map {H_{n + 1} } {\dfrac 1 {n + 1}, \dfrac 1 {n + 1}, \dotsc, \dfrac 1 {n + 1} }$
Thus, for example, a $2$-horse race is less uncertain than a $3$-horse race.
Sources
- 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty