This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Probabilistic metric space" – news ·newspapers ·books ·scholar ·JSTOR(July 2023) (Learn how and when to remove this message)

Inmathematics,probabilistic metric spaces are a generalization ofmetric spaces where thedistance no longer takes values in the non-negativereal numbersR_≥ 0, but in distribution functions.^[1]

LetD+ be the set of allprobability distribution functionsF such thatF(0) = 0 (F is a nondecreasing, leftcontinuous mapping fromR into [0, 1] such thatmax(F) = 1).

Then given anon-empty setS and a functionF:S ×S →D+ where we denoteF(p, q) byF_p,q for every (p, q) ∈S ×S, theordered pair (S, F) is said to be a probabilistic metric space if:

• For allu andv inS,u =v if and only ifF_u,v(x) = 1 for allx > 0.
• For allu andv inS,F_u,v =F_v,u.
• For allu,v andw inS,F_u,v(x) = 1 andF_v,w(y) = 1 ⇒F_u,w(x +y) = 1 forx,y > 0.^[2]

Probabilistic metric spaces are initially introduced by Menger, which were termedstatistical metrics.^[3] Shortly after, Wald criticized the generalizedtriangle inequality and proposed an alternative one.^[4] However, both authors had come to the conclusion that in some respects the Wald inequality was too stringent a requirement to impose on all probability metric spaces, which is partly included in the work of Schweizer and Sklar.^[5] Later, the probabilistic metric spaces found to be very suitable to be used with fuzzy sets^[6] and further called fuzzy metric spaces^[7]

A probability metricD between tworandom variablesX andY may be defined, for example, as$${\displaystyle D(X,Y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|F(x,y)dxdy}$$whereF(x,y) denotes the joint probability density function of the random variablesX andY. IfX andY are independent from each other, then the equation above transforms into$${\displaystyle D(X,Y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|f(x)g(y)dxdy}$$wheref(x) andg(y) are probability density functions ofX andY respectively.

One may easily show that such probability metrics do not satisfy the firstmetric axiom or satisfies itif, and only if, both of argumentsX andY are certain events described byDirac delta densityprobability distribution functions. In this case:$${\displaystyle D(X,Y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x-y|\delta (x-{\mu }_x)\delta (y-{\mu }_y)dxdy=|{\mu }_x-{\mu }_y|}$$the probability metric simply transforms into the metric betweenexpected values${\displaystyle {\mu }_x}$,${\displaystyle {\mu }_y}$ of the variablesX andY.

For all otherrandom variablesX,Y the probability metric does not satisfy theidentity of indiscernibles condition required to be satisfied by the metric of the metric space, that is:$${\displaystyle D\left(X,X\right)>0.}$$

For example if bothprobability distribution functions of random variablesX andY arenormal distributions (N) having the samestandard deviation${\displaystyle \sigma }$, integrating${\displaystyle D\left(X,Y\right)}$ yields:$${\displaystyle D_{NN}(X,Y)={\mu }_{xy}+\frac{2\sigma }{\sqrt{\pi }}\exp \left(-\frac{{\mu }_{xy}^2}{4{\sigma }^2}\right)-{\mu }_{xy}\mathrm{erfc}\left(\frac{{\mu }_{xy}}{2\sigma }\right)}$$where$${\displaystyle {\mu }_{xy}=\left|{\mu }_x-{\mu }_y\right|,}$$and${\displaystyle \mathrm{erfc}(x)}$ is the complementaryerror function.

In this case:$${\displaystyle \underset{{\mu }_{xy}\to 0}{lim}{D}_{NN}(X,Y)=D_{NN}(X,X)=\frac{2\sigma }{\sqrt{\pi }}.}$$

The probability metric of random variables may be extended into metricD(X,Y) ofrandom vectorsX,Y by substituting${\displaystyle |x-y|}$ with any metric operatord(x,y):$${\displaystyle D(\mathbf{X},\mathbf{Y})={\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}d(\mathbf{x},\mathbf{y})F(\mathbf{x},\mathbf{y})d{\mathrm{\Omega }}_{x}d{\mathrm{\Omega }}_y}$$whereF(X,Y) is the joint probability density function of random vectorsX andY. For example substitutingd(x,y) withEuclidean metric and providing the vectorsX andY are mutually independent would yield to:$${\displaystyle D(\mathbf{X},\mathbf{Y})={\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\sqrt{\sum _i|x_i-y_i{|}^2}F(\mathbf{x})G(\mathbf{y})d{\mathrm{\Omega }}_{x}d{\mathrm{\Omega }}_y.}$$

• Probability distributions
• Metric geometry

• Articles needing additional references from July 2023
• All articles needing additional references