Fréchet principle
The Fréchet principle , named after the French mathematician Maurice René Fréchet (1878–1973), is a general recipe for defining the expected value and variance for random variables in a suitable manner . The principle goes back to a work by Fréchet from 1948.
Definitions
Let be a metric space , where a metric is on the basic set . Let further be a random variable with values in . Then the Fréchet expectation value is defined as the value that minimizes the expected square metric distance between and the elements , i.e. H.
- .
This minimum achievable expected square distance defines the variance, ie
- .
Examples
- For the Euclidean metric and a classical random variable fulfill and the Fréchet principle, d. H. it applies
- .
- It is a bit more complicated in the case of random sets or random fuzzy sets . The Hausdorff metric is often used there as a metric , but the i. General did not use Aumann expectation as Fréchet expectation with respect to . This is not suitable for meaningfully defining a variance for random (fuzzy) sets. At least for random convex (fuzzy) sets, however, the Fréchet expectation value with respect to a metric defined by the carrier functions of convex sets. This then naturally gives the variance of a random convex (fuzzy) set.