Random amount

A random set is a set whose characteristics (e.g. size, shape, location) also depend on chance, e.g. B. the spatiotemporal development of an epidemic or an oil spill on the ocean. Random sets are also fundamental to stochastic geometry .

definition

A random set is a set valued random variable ; H. a measurable mapping from a probability space into a measurable space . Often the set of all compact subsets of a locally compact separable Hausdorff space and the sigma algebra generated by . Then one speaks of a random compact set , see e.g. B. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {B}},P)}$ ${\displaystyle (K,{\mathcal {B}}(K))}$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {B}}(K)}$ ${\displaystyle K}$

Distribution of a random compact set

Be a random compact set. The distribution of is uniquely determined by the probabilities with which any one of "hits" (so-called hit probabilities ), i. H. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle K}$

${\displaystyle P(X\cap A\neq \emptyset )=T(A),\quad A\in K}$

${\displaystyle T(A)}$is a fully alternating capacity .

Expected value of a random compact set

Be a random compact set. Their expected value is often called the Aumann expected value . It is defined as the set of all expectation values of random variables that are almost certainly in , i.e. H. ${\displaystyle X}$ ${\displaystyle E_{A}X}$ ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle E_{A}X=\{EY:Y\in X\quad f.s.\}}$.

They are also called selectors of . For a random interval, z. B. ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle E_{A}[a,b]=[Ea,Eb]}$.

The Aumann expectation value is linear with respect to the Minkowski sum , i.e. H. ${\displaystyle \oplus }$

${\displaystyle E_{A}(X_{1}\oplus X_{2})=E_{A}X_{1}\oplus E_{A}X_{2}}$.

Bibliography

• Matheron, G. (1975) Random Sets and Integral Geometry . J. Wiley & Sons, New York.
• Molchanov, I. (2005) The Theory of Random Sets . Springer, New York.
• Stoyan D., and H. Stoyan (1994) Fractals, Random Shapes and Point Fields . John Wiley & Sons, Chichester, New York.

Individual evidence

1. ^ Matheron, G. (1975) Random Sets and Integral Geometry . J. Wiley & Sons, New York.
2. ↑ Aumann, J. (1965). Integral of set valued functions . Journ.Math.Anal.Appl.12, 1-22.