Probabilistic metric space

A probability metric D between two random variables X and Y may be defined, for example, as $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|F(x,y)\,dx\,dy}$$ where F(x, y) denotes the joint probability density function of the random variables X and Y. If X and Y are independent from each other, then the equation above transforms into $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|f(x)g(y)\,dx\,dy}$$ 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 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: $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|\delta (x-\mu _{x})\delta (y-\mu _{y})\,dx\,dy=|\mu _{x}-\mu _{y}|}$$ the probability metric simply transforms into the metric between expected values ${\displaystyle \mu _{x}}$, ${\displaystyle \mu _{y}}$ 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: $${\displaystyle D\left(X,X\right)>0.}$$

For example if both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation ${\displaystyle \sigma }$, integrating ${\displaystyle D\left(X,Y\right)}$ yields: $${\displaystyle D_{NN}(X,Y)=\mu _{xy}+{\frac {2\sigma }{\sqrt {\pi }}}\exp \left(-{\frac {\mu _{xy}^{2}}{4\sigma ^{2}}}\right)-\mu _{xy}\operatorname {erfc} \left({\frac {\mu _{xy}}{2\sigma }}\right)}$$ where $${\displaystyle \mu _{xy}=\left|\mu _{x}-\mu _{y}\right|,}$$ and ${\displaystyle \operatorname {erfc} (x)}$ is the complementary error function.

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

The probability metric of random variables may be extended into metric D(X, Y) of random vectors X, Y by substituting ${\displaystyle |x-y|}$ with any metric operator d(x, y): $${\displaystyle D(\mathbf {X} ,\mathbf {Y} )=\int _{\Omega }\int _{\Omega }d(\mathbf {x} ,\mathbf {y} )F(\mathbf {x} ,\mathbf {y} )\,d\Omega _{x}d\Omega _{y}}$$ where F(X, Y) is the joint probability density function of random vectors X and Y. For example substituting d(x, y) with Euclidean metric and providing the vectors X and Y are mutually independent would yield to: $${\displaystyle D(\mathbf {X} ,\mathbf {Y} )=\int _{\Omega }\int _{\Omega }{\sqrt {\sum _{i}|x_{i}-y_{i}|^{2}}}F(\mathbf {x} )G(\mathbf {y} )\,d\Omega _{x}d\Omega _{y}.}$$