Independent quantity systems

Independent set systems are considered in probability theory , a branch of mathematics . The independence of set systems is a generalization of the stochastic independence of events and is used to define the stochastic independence of random variables . Thus, independent set systems belong to the basic concepts of stochastics and are a building block for many requirements of important theorems of statistics and stochastics.

definition

A probability space is given , i.e. a σ-algebra on the basic set and a probability measure . Furthermore, let an arbitrary index set and for each index a set system be given. ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle P\colon {\mathcal {A}}\to [0;1]}$ ${\displaystyle I}$ ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {E}}_{i}\subseteq {\mathcal {A}}}$

The family of set systems is now called independent if and only if for every finite subset and every possible choice of events with these events are stochastically independent , that is, if applies in each case ${\displaystyle ({\mathcal {E}}_{i})_{i\in I}}$ ${\displaystyle J\subset I}$ ${\displaystyle (E_{j})_{j\in J}}$ ${\displaystyle \forall j\in J\colon E_{j}\in {\mathcal {E}}_{j}}$

${\displaystyle P\left(\bigcap _{j\in J}E_{j}\right)=\prod _{j\in J}P(E_{j})}$.

Examples

• If and , then the set systems are independent if and only if the two events and are independent. It is , therefore, the cases and review. The case is trivial. ${\displaystyle {\mathcal {E}}_{1}=\{A\}}$ ${\displaystyle {\mathcal {E}}_{2}=\{B\}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle I=\{1,2\}}$ ${\displaystyle J=\{1\},J=\{2\}}$ ${\displaystyle J=I=\{1,2\}}$ ${\displaystyle J=\emptyset }$

1. Is , so is always , because the system of sets is one-element. So the statement is always true. The case follows analogously . ${\displaystyle J=\{1\}}$ ${\displaystyle E_{1}=A}$ ${\displaystyle P\left(\bigcap _{j\in \{1\}}E_{j}\right)=P(A)=\prod _{j\in \{1\}}P(E_{j})}$ ${\displaystyle J=\{2\}}$
2. Is , then again taking advantage of the unity of the set systems ( ) ${\displaystyle J=I=\{1,2\}}$ ${\displaystyle E_{1}=A,E_{2}=B}$

${\displaystyle P(E_{1}\cap E_{2})=P(A\cap B)=P(A)P(B)=P(E_{1})P(E_{2})}$

due to the independence of and . ${\displaystyle A}$ ${\displaystyle B}$

• If, more generally, a family of events and if the family of set systems is defined as one-element set systems by for all , then the family of set systems is independent if and only if the family is independent of events. This equivalence is also used in part to define independence from events. ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle {\mathcal {E}}_{i}=\{A_{i}\}}$ ${\displaystyle i\in I}$
• A σ-algebra on a probability space is called a P-trivial σ-algebra if either or holds for all . P-trivial σ-algebras are independent of any system of sets. Because is and , so is for anything from another system of sets . The same applies if is. So are and independent. ${\displaystyle {\mathcal {O}}}$ ${\displaystyle A\in {\mathcal {O}}}$ ${\displaystyle P(A)=0}$ ${\displaystyle P(A)=1}$ ${\displaystyle A\in {\mathcal {O}}}$ ${\displaystyle P(A)=0}$ ${\displaystyle P(A\cap B)=0=P(A)\cdot P(B)}$ ${\displaystyle B}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle P(A\cap B)=1\cdot P(B)}$ ${\displaystyle P(A)=1}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {M}}}$

properties

• If is a disjoint decomposition of (that is, it is for all and it is ) and if the family of systems of sets is independent, then the family of systems of sets is defined by ${\displaystyle (I_{k})_{k\in K}}$ ${\displaystyle I}$ ${\displaystyle I_{k'}\cap I_{k^{*}}=\emptyset }$ ${\displaystyle k',k^{*}\in K,k'\neq k^{*}}$ ${\displaystyle \bigcup _{k\in K}I_{k}=I}$ ${\displaystyle ({\mathcal {E}}_{i})_{i\in I}}$

${\displaystyle \left(\bigcup _{i\in I_{k}}{\mathcal {E}}_{i}\right)_{k\in K}}$

independently.

• For finite things the following applies: If each of the set systems already contains the superset , then they are independent if and only if ${\displaystyle I}$ ${\displaystyle \Omega }$

${\displaystyle P\left(\bigcap _{i\in I}E_{i}\right)=\prod _{i\in I}P(E_{j})}$

for everyone . It is then sufficient to check the defining equation only for the entire index set. For the equation follows automatically if you bet forever . ${\displaystyle E_{i}\in {\mathcal {E}}_{i}}$ ${\displaystyle J\subset I}$ ${\displaystyle i\in I\setminus J}$ ${\displaystyle E_{i}=\Omega }$

• If the set system for each is an average stable set system, then is independent if and only if the generated σ-algebras are independent. ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {E}}_{i}\cup \{\emptyset \}}$ ${\displaystyle ({\mathcal {E}}_{i})_{i\in I}}$ ${\displaystyle (\sigma ({\mathcal {E}}_{i}))_{i\in I}}$

use

Independent set systems are used to transfer independence to random variables. Be a probability space and two measuring rooms and two random variables of for or given. If the two initial σ-algebras generated by the random variables are independent set systems, then the random variables are called independent . This can also be generalized to families of random variables. ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1}),(\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle X_{1},X_{2}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$

Independence from random variables and set systems

In the context of the conditional expected value , the independence of a random variable and a set system is sometimes also used . The random variable and the set system are said to be independent if the set system and the initial σ-algebra of the random variable are independent set systems in the above sense. ${\displaystyle X}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle \sigma (X)}$

generalization

The independence of σ-algebras can be extended to the conditional independence by means of the conditional expectation value . It also exists for random variables.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .