In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure. [1]
Note the difference in notation between the expectation value of a random element , denoted by and the intensity measure of the random measure , denoted by .
Propertiesedit
The intensity measure is always s-finite and satisfies
intensity, measure, probability, theory, intensity, measure, measure, that, derived, from, random, measure, intensity, measure, random, measure, defined, expectation, value, random, measure, hence, corresponds, average, volume, random, measure, assigns, intens. In probability theory an intensity measure is a measure that is derived from a random measure The intensity measure is a non random measure and is defined as the expectation value of the random measure of a set hence it corresponds to the average volume the random measure assigns to a set The intensity measure contains important information about the properties of the random measure A Poisson point process interpreted as a random measure is for example uniquely determined by its intensity measure 1 Definition editLet z displaystyle zeta nbsp be a random measure on the measurable space S A displaystyle S mathcal A nbsp and denote the expected value of a random element Y displaystyle Y nbsp with E Y displaystyle operatorname E Y nbsp The intensity measure E z A 0 displaystyle operatorname E zeta colon mathcal A to 0 infty nbsp of z displaystyle zeta nbsp is defined as E z A E z A displaystyle operatorname E zeta A operatorname E zeta A nbsp for all A A displaystyle A in mathcal A nbsp 2 3 Note the difference in notation between the expectation value of a random element Y displaystyle Y nbsp denoted by E Y displaystyle operatorname E Y nbsp and the intensity measure of the random measure z displaystyle zeta nbsp denoted by E z displaystyle operatorname E zeta nbsp Properties editThe intensity measure E z displaystyle operatorname E zeta nbsp is always s finite and satisfies E f x z d x f x E z d x displaystyle operatorname E left int f x zeta mathrm d x right int f x operatorname E zeta dx nbsp for every positive measurable function f displaystyle f nbsp on S A displaystyle S mathcal A nbsp 3 References edit Klenke Achim 2008 Probability Theory Berlin Springer p 528 doi 10 1007 978 1 84800 048 3 ISBN 978 1 84800 047 6 Klenke Achim 2008 Probability Theory Berlin Springer p 526 doi 10 1007 978 1 84800 048 3 ISBN 978 1 84800 047 6 a b Kallenberg Olav 2017 Random Measures Theory and Applications Switzerland Springer p 53 doi 10 1007 978 3 319 41598 7 ISBN 978 3 319 41596 3 Retrieved from https en wikipedia org w index php title Intensity measure amp oldid 1061676741, wikipedia, wiki, book, books, library,