Axiom:Axioms of Uncertainty/Axiom 3
Jump to navigation
Jump to search
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 $Z$.
$H$ fulfils the following axiom:
- $\map {H_n} {p_1, p_2, \ldots, p_n} \ge 0$
and:
- $\map {H_n} {p_1, p_2, \ldots, p_n} = 0$ if and only if $\exists i \in \set {1, 2, \ldots, n}: p_i = 1$
That is, uncertainty is always a positive quantity, and only reaches zero when one of the arguments is $1$.
Sources
- 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty