Probabilistic metric space

A probabilistic metric space is a generalization of metric spaces where the distance is no longer defined on positive real numbers, but on distribution functions.

Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that sup(F(x)) = 1 for x∈R.

The ordered pair (S,F) is said to be a probabilistic metric space if S is a nonempty set and F: S×S → D+ (F(p, q) is denoted by F_p,q for every (p, q) ∈ S × S) satisfies the following conditions:

A probability metric D between two random variables X and Y may be defined e.g. as:

where F(x, y) 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(x) and g(y) 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 is only if, and only if, both of its arguments X, 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 ${\displaystyle \mu _{x}}$, ${\displaystyle \mu _{y}}$ of the variables X and Y.

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