Uncertainty Function satisfies Axioms of Uncertainty

Theorem

Let $X$ be a random variable.

Let $X$ take a finite number of values with probabilities $p_1, p_2, \dotsc, p_n$.

Let $\map H X$ be the uncertainty function of $X$:

$\map H X = \displaystyle -\sum_k p_k \lg p_k$

where:

$\lg$ denotes logarithm base $2$

the summation is over those $k$ where $p_k > 0$.

Then the uncertainty function satisfies the Axioms of Uncertainty.

Sources

• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty: Exercises $1.1$: $2.$