Probabilistic metric space


A probabilistic metric space is a generalization of metric spaces where the distance has no longer values in non-negative real numbers, but in distribution functions.
Let D+ be the set of all probability distribution functions F such that F = 0.
The ordered pair is said to be a probabilistic metric space if S is a nonempty set and F: S×S →
D+ is denoted by Fp,q for every satisfies the following conditions:
A probability metric D between two random variables X and Y may be defined e.g. as:
where F denotes the joint probability density function of random variables X and Y. Obviously if X and Y are independent from each other the equation above transforms into:
where f and g are probability density functions of X and Y respectively.
One may easily show that such probability metrics do not satisfy the first metric axiom or satisfies it if, and only if, both of arguments X and Y are certain events described by Dirac delta density probability distribution functions. In this case:
the probability metric simply transforms into the metric between expected values, of the variables X and Y.
For all other random variables X, Y the probability metric does not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space, that is:
s and the same standard deviation .
denotes a distance between means of X and Y.

Example

For example if both probability distribution functions of random variables X and Y are normal distributions having the same standard deviation, integrating yields to:
where:
and is the complementary error function.
In this case:

Probability metric of random vectors

The probability metric of random variables may be extended into metric D of random vectors X, Y by substituting with any metric operator d:
where F is the joint probability density function of random vectors X and Y.
For example substituting d with Euclidean metric and providing the vectors X and Y are mutually independent would yield to: