Empirical Distribution (Random Measure)
The empirical distribution is a random measure in stochastics , a branch of mathematics . It forms a discrete probability distribution , the exact structure of which depends on several random variables . Only at the transition to realizations of the random variables is the probability distribution clearly determined and then the empirical distribution of a sample . The empirical distribution plays an important role in the area between probability theory and statistics . For example, the expected value of the empirical distribution can be used under certain circumstances as an estimate function for the expected value of the underlying random variable.
definition
Real random variables are given .
Then is called
the empirical distribution of . If all random variables have the same distribution, sometimes only the sample size and one random variable are given.
The empirical distribution is also defined somewhat more generally for random variables with values in Polish spaces .
Demarcation
The discrete probability distribution is also used as an empirical distribution
marked with . This is referred to here as the empirical distribution of the sample . The empirical distribution of the random variables and the empirical distribution of the sample are closely related. The empirical distribution of the sample arises when one moves from the (indefinite) random variable to the realization of the random variable.
properties
Expected value
The expected value of the empirical distribution is the sample mean , that is
Median
The median of the empirical distribution is the sample median , that is
- .
Here the i-th order statistic denotes .
Variance
The variance of the empirical distribution is the (uncorrected) sample variance , so
Moments
The kth moment of the empirical distribution is given by
- .
The moments of the empirical distribution are also known as the sampling moment .
use
If the random variables are independently and identically distributed , the key figures of the empirical distribution can serve as an estimation function for the corresponding key figures of the random variables . The sample mean is the expected value of the empirical distribution and can be used as an estimator for the expected value of the random variable .
Web links
- AV Prokhorov: Empirical distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Individual evidence
- ↑ a b Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 245 , doi : 10.1007 / 978-3-642-36018-3 .
- ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 243 , doi : 10.1515 / 9783110215274 .
- ↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 357 , doi : 10.1007 / 978-3-642-21026-6 .
- ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 75 , doi : 10.1007 / 978-3-642-17261-8 .