Liouville's theorem (Hamiltonian): Difference between revisions

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{{Probability fundamentals}}
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In [[probability theory]], the '''law''' (or '''formula''') '''of total probability''' is a fundamental rule relating [[Marginal probability|marginal probabilities]] to [[conditional probabilities]]. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name.
 
==Statement==
The law of total probability is<ref name= ZK>Zwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. ISBN 1-58488-059-7 page 31.</ref> the [[proposition]] that if <math>\left\{{B_n : n = 1, 2, 3, \ldots}\right\}</math> is a finite or [[Countable set|countably infinite]] [[partition of a set|partition]] of a [[sample space]] (in other words, a set of [[pairwise disjoint]] [[Event (probability theory)|event]]s whose [[union (set theory)|union]] is the entire sample space) and each event <math>B_n</math> is [[measurable set|measurable]], then for any event <math>A</math> of the same [[probability space]]:
 
:<math>\Pr(A)=\sum_n \Pr(A\cap B_n)\,</math>
 
or, alternatively,<ref name=ZK/>
 
:<math>\Pr(A)=\sum_n \Pr(A\mid B_n)\Pr(B_n),\,</math>
 
where, for any <math>n\,</math> for which <math>\Pr(B_n) = 0 \,</math> these terms are simply omitted from the summation, because  <math>\Pr(A\mid B_n)\,</math> is finite.
 
The summation can be interpreted as a [[weighted average]], and consequently the marginal probability, <math>\Pr(A)</math>, is sometimes called "average probability";<ref name="Pfeiffer1978">{{cite book|author=Paul E. Pfeiffer|title=Concepts of probability theory|url=http://books.google.com/books?id=_mayRBczVRwC&pg=PA47|year=1978|publisher=Courier Dover Publications|isbn=978-0-486-63677-1|pages=47–48}}</ref> "overall probability" is sometimes used in less formal writings.<ref name="Rumsey2006">{{cite book|author=Deborah Rumsey|title=Probability for dummies|url=http://books.google.com/books?id=Vj3NZ59ZcnoC&pg=PA58|year=2006|publisher=For Dummies|isbn=978-0-471-75141-0|page=58}}</ref>
 
The law of total probability can also be stated for conditional probabilities. Taking the <math>B_n</math> as above, and assuming <math>C</math> is an event [[Independence (probability theory)|independent]] with any of the <math>B_n</math>:
 
:<math>\Pr(A \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n) </math>
 
==Informal formulation==
The above mathematical statement might be interpreted as follows: ''given an outcome <math>A</math>, with known conditional probabilities given any of the <math>B_n</math> events, each with a known probability itself, what is the total probability that <math>A</math> will happen?''. The answer to this question is given by <math>\Pr(A)</math>.
 
==Example==
Suppose that two factories supply [[light bulb]]s to the market. Factory ''X''<nowiki>'</nowiki>s bulbs work for over 5000 hours in 99% of cases, whereas factory ''Y''<nowiki>'</nowiki>s bulbs work for over 5000 hours in 95% of cases. It is known that factory ''X'' supplies 60% of the total bulbs available. What is the chance that a purchased bulb will work for longer than 5000 hours?
 
Applying the law of total probability, we have:
 
<math>{\Pr(A)=\Pr(A|B_1)}\cdot{\Pr(B_1)}+{\Pr(A|B_2)}\cdot{\Pr(B_2)}={99 \over 100}\cdot{6 \over 10}+{95 \over 100}\cdot{4 \over 10}={{594 + 380} \over 1000}={974 \over 1000}</math>
 
where
* <math>\Pr(B_1)={6 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''X'';
* <math>\Pr(B_2)={4 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''Y'';
* <math>\Pr(A|B_1)={99 \over 100}</math> is the probability that a bulb manufactured by ''X'' will work for over 5000h;
* <math>\Pr(A|B_2)={95 \over 100}</math> is the probability that a bulb manufactured by ''Y'' will work for over 5000h.
 
Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.
 
==Applications==
One common application of the law is where the events coincide with a [[discrete random variable]] ''X'' taking each value in its range, i.e. <math>B_n</math> is the event <math>X=x_n</math>. It follows that the probability of an event ''A'' is equal to the [[expected value]] of the [[conditional probability|conditional probabilities]] of ''A'' given <math>X=x_n</math>.{{Citation needed|date=September 2010}}  That is,
 
:<math>\Pr(A)=\sum_n \Pr(A\mid X=x_n)\Pr(X=x_n) = \operatorname{E}[\Pr(A\mid X)] ,</math>
 
where Pr(''A''&nbsp;|&nbsp;''X'') is the [[conditional probability]] of ''A'' given the value of the random variable ''X''.<ref name="Rumsey2006" />  This conditional probability is a random variable in its own right, whose value depends on that of&nbsp;''X''. The conditional probability Pr(''A''&nbsp;|&nbsp;''X''&nbsp;=&nbsp;x) is simply a conditional probability given an event, [''X''&nbsp;=&nbsp;''x''].  It is a function of ''x'', say ''g''(''x'')&nbsp;=&nbsp;Pr(''A''&nbsp;|&nbsp;''X''&nbsp;=&nbsp;''x'').  Then the conditional probability Pr(''A''&nbsp;|&nbsp;''X'') is ''g''(''X''), hence itself a random variable.  This version of the law of total probability says that the expected value of this random variable is the same as&nbsp;Pr(''A'').
 
This result can be generalized to [[continuous random variable]]s (via [[continuous conditional density]]), and the expression becomes
 
:<math>\Pr(A)= \operatorname{E}[\Pr(A\mid \mathcal{F}_X)],</math>
 
where <math>\mathcal{F}_X</math> denotes the [[sigma-algebra]] generated by the random variable ''X''.{{Citation needed|date=November 2010}}
 
==Other names==
The term '''''law of total probability''''' is sometimes taken to mean the '''law of alternatives''', which is a special case of the law of total probability applying to [[discrete random variable]]s.{{Citation needed|date=September 2010}} One author even uses the terminology "continuous law of alternatives" in the continuous case.<ref name="Baclawski2008">{{cite book|author=Kenneth Baclawski|title=Introduction to probability with R|url=http://books.google.com/books?id=Kglc9g5IPf4C&pg=PA179|year=2008|publisher=CRC Press|isbn=978-1-4200-6521-3|page=179}}</ref> This result is given by Grimmett and Welsh<ref>''Probability: An Introduction'', by [[Geoffrey Grimmett]] and [[Dominic Welsh]], Oxford Science Publications, 1986, Theorem 1B.</ref> as the '''partition theorem''', a name that they also give to the related [[law of total expectation]].
 
==See also==
* [[Law of total expectation]]
* [[Law of total variance]]
* [[Law of total cumulance]]
 
== References ==
 
<references/>
 
* ''Introduction to Probability and Statistics'' by William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
* ''Theory of Statistics'', by Mark J. Schervish, Springer, 1995.
* ''Schaum's Outline of Theory and Problems of Beginning Finite Mathematics'', by John J. Schiller, Seymour Lipschutz, and R. Alu Srinivasan, McGraw–Hill Professional, 2005, page 116.
* ''A First Course in Stochastic Models'', by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
* ''An Intermediate Course in Probability'', by Alan Gut, Springer, 1995, pages 5–6.
 
{{DEFAULTSORT:Law Of Total Probability}}
[[Category:Probability theorems]]
[[Category:Statistical laws]]

Latest revision as of 14:38, 10 January 2015

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