Appendix A The Basic Formalism of Probability Theory

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K3 Any countable sequence of mutually disjoint events E 1 , E 2 , satis es P (E 1 E 2 ) = P (E 1 ) + P (E 2 ) + (A7)

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That is, the probability of an event set which is the union of other disjoint subsets is the sum of the probabilities of those subsets This is called -additivity If there is any overlap among the subsets, this relation does not hold A probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satis ed In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals

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A6 Lemmas in Probability

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(i) From the Kolmogorov axioms one deduces other useful rules for calculating probabilities: P (A B) = P (A) + P (B) P (A B) (A8)

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That is, the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that A and B will happen This can be extended to the inclusion exclusion principle (ii) P ( E ) = 1 P (E ) (A9) That is, the probability that any event will not happen is 1 minus the probability that it will (iii) Using conditional probability as de ned above, it also follows immediately that P (A B) = P (A) P (B|A) (A10)

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That is, the probability that A and B will happen is the probability that A will happen, times the probability that B will happen given that A happened; this relationship gives Bayes s theorem (see next) It then follows that A and B are independent if and only if P (A B) = P (A) P (B) (A11)

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A7 Bayes s Theorem

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Bayes s theorem relates the conditional and marginal probabilities of events A and B, where B has a nonvanishing probability: P (A|B) = P (B|A) P (A) P (B) (A12)

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A8 Random Variable

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More generally, let = space Then we have P (Ai |B) =

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Ai , Ai A j = for i = j , be a partition of the event /

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P (B|Ai ) P (Ai ) , j P (B|A j ) P (A j )

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(A13)

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for any Ai in the partition

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A8 Random Variable

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Let ( , F , P ) be a probability space and (Y , ) be a measurable space Then a random variable is formally de ned as a measurable function : Y In other words, the preimage of the well-behaved subsets of Y (the elements of ) are events, and hence are assigned a probability by P When the measurable space is the measurable space over the real numbers, one speaks of real-valued random variables Then, the function is a real-valued random variable if { | ( ) u r} F , r R (A14)

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A9 Probability Distribution

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The probability distribution of a real-valued variable can be uniquely described by its cumulative distribution function F (x) (also called a distribution function), which is de ned by F (x) = P ( u x) (A15)

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for any x in R More generally, given a random variable : Y between a probability space ( , F , P ), the sample space, and a measurable space (Y , ), called the state space, a probability distribution on (Y , ) is a probability measure P : [0, 1] on the state space, where P is the push-forward measure of P A distribution is called discrete if its cumulative distribution function consists of a sequence of nite jumps, which means that it belongs to a discrete random variable , a variable which can only attain values from a certain nite or countable set Discrete distributions are characterized by a probability mass function p such that P ( = x) = p(x) A distribution is called continuous if its cumulative distribution function is continuous A random variable is called continuous if its distribution function is continuous In that case P ( = x) = 0 for all x R Note also that there are probability distribution functions which are neither discrete nor continuous

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