# Axiom:Axioms of Uncertainty/Axiom 8

## 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:

Let:

$p = p_1 + p_2 + \dotsb + p_m$
$q = q_1 + q_2 + \dotsb + q_n$

such that:

each of $p_i$ and $q_j$ are non-negative
$p + q = 1$

Then:

$\map {H_{m + n} } {p_1, p_2, \dotsc, p_m, q_1, q_2, \dotsc q_n} = \map {H_2} {p, q} + p \map {H_m} {\dfrac {p_1} p, \dfrac {p_2} p, \dotsc, \dfrac {p_m} p} + q \map {H_n} {\dfrac {q_1} q, \dfrac {q_2} q, \dotsc, \dfrac {q_n} q}$