# Entropy of a probability distribution

The **entropy** of a probability distribution is a number that describes the degree of uncertainty or disorder the distribution represents.

## Examples

Assume we have a set of two mutually exclusive propositions (or equivalently, a random experiment with two possible outcomes). Assume all two possibilities are equally likely.

Then our advance uncertainty about the eventual outcome is rather small - we know in advance it will be one of exactly two known alternatives.

Assume now we have a set of a million alternatives - all of them equally likely - rather than two.

It seems clear that our uncertainty now about the eventual outcome will be much bigger.

## Formal definitions

- Given a discrete probability distribution function f, the entropy H of the distribution (measured in bits) is given by <math>H=-\sum_{\forall i : f(x_i) \ne 0}^{} f(x_{i}) log_{2} f(x_{i} )</math>
- Given a continuous probability distribution function f, the entropy H of the distribution (again measure in bits) is given by <math>H=-\int_{\ x: f(x) \ne 0 } f(x) log_{2} f(x) dx</math>

Note that some authors prefer to use other units than bit to measure entropy, the formulas are then slightly different. Also, the symbol S is sometimes used, rather than H.

## See also

- Entropy in thermodynamics and information theory
- Discrete probability distribution
- Continuous probability distribution
- Probability
- Probability theory