Function that Satisfies Axioms of Uncertainty

Theorem
Let $Z$ be a random variable.

Let $n \in \N$ be a natural number.

Let $p_1, p_2, \dotsc, p_n$ be real numbers such that:
 * $\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$
 * $\displaystyle \sum_{i \mathop = 1}^n p_i = 1$

Let $\map H {p_1, p_2, \ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty.

Then:
 * $\map H {p_1, p_2, \ldots, p_n} = \displaystyle -\lambda \sum_{i \mathop = 1}^n p_i \log_b p_i$

where $b \in \R_{>1}$.

Thus the uncertainty function satisfies these axioms.