Łukaszyk–Karmowski metric

In mathematics, the Łukaszyk–Karmowski metric is a function defining a distance between two random variables or two random vectors. This function is not a metric as it does not satisfy the identity of indiscernibles condition of the metric, that is for two identical arguments its value is greater than zero. The concept is named after Szymon Łukaszyk and Wojciech Karmowski.

The Łukaszyk–Karmowski metric D between two continuous independent random variables X and Y is defined as:

where f(x) and g(y) are the probability density functions of X and Y respectively.

One may easily show that such metrics above do not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space. In fact they satisfy this condition if and only if both arguments X, Y are certain events described by Dirac delta density probability distribution functions. In such a case:

the Łukaszyk–Karmowski metric simply transforms into the metric between expected values ${\displaystyle \mu _{x}}$, ${\displaystyle \mu _{y}}$ of the variables X and Y and obviously:

...
Wikipedia