resources_Cheatsheet.pdf - Probability\u2013the Science of Uncertainty and Data by Fabi\u00b4 an Kozynski Theorem(Bayes\u2019 rule Given a partition{A1 A2 of the

resources_Cheatsheet.pdf - Probability–the Science of...

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Probability–the Science of Uncertainty and Data by Fabi´an Kozynski Probability Probability models and axioms Definition (Sample space) A sample space Ω is the set of all possible outcomes. The set’s elements must be mutually exclusive, collectively exhaustive and at the right granularity. Definition (Event) An event is a subset of the sample space. Probability is assigned to events. Definition (Probability axioms) A probability law P assigns probabilities to events and satisfies the following axioms: Nonnegativity P ( A ) ≥ 0 for all events A . Normalization P ( Ω ) = 1. (Countable) additivity For every sequence of events A 1 , A 2 , . . . such that A i A j = ∅ : P ( i A i ) = i P ( A i ) . Corollaries (Consequences of the axioms) P (∅) = 0. For any finite collection of disjoint events A 1 , . . . , A n , P ( n i = 1 A i ) = n i = 1 P ( A i ) . P ( A ) + P ( A c ) = 1. P ( A ) ≤ 1. If A B , then P ( A ) ≤ P ( B ) . P ( A B ) = P ( A ) + P ( B ) − P ( A B ) . P ( A B ) ≤ P ( A ) + P ( B ) . Example (Discrete uniform law) Assume Ω is finite and consists of n equally likely elements. Also, assume that A Ω with k elements. Then P ( A ) = k n . Conditioning and Bayes’ rule Definition (Conditional probability) Given that event B has occurred and that P ( B ) > 0, the probability that A occurs is P ( A B ) = P ( A B ) P ( B ) . Remark (Conditional probabilities properties) They are the same as ordinary probabilities. Assuming P ( B ) > 0: P ( A B ) ≥ 0. P ( Ω B ) = 1 P ( B B ) = 1. If A C = ∅ , P ( A C B ) = P ( A B ) + P ( C B ) . Proposition (Multiplication rule) P ( A 1 A 2 ∩⋯∩ A n ) = P ( A 1 )⋅ P ( A 2 A 1 )⋯ P ( A n A 1 A 2 ∩⋯∩ A n 1 ) . Theorem (Total probability theorem) Given a partition { A 1 , A 2 , . . . } of the sample space, meaning that i A i = Ω and the events are disjoint, and for every event B , we have P ( B ) = i P ( A i ) P ( B A i ) . Theorem (Bayes’ rule) Given a partition { A 1 , A 2 , . . . } of the sample space, meaning that i A i = Ω and the events are disjoint, and if P ( A i ) > 0 for all i , then for every event B , the conditional probabilities P ( A i B ) can be obtained from the conditional probabilities P ( B A i ) and the initial probabilities P ( A i ) as follows: P ( A i B ) = P ( A i ) P ( B A i ) j P ( A j ) P ( B A j ) . Independence Definition (Independence of events) Two events are independent if occurrence of one provides no information about the other. We say that A and B are independent if P ( A B ) = P ( A ) P ( B ) . Equivalently, as long as P ( A ) > 0 and P ( B ) > 0, P ( B A ) = P ( B ) P ( A B ) = P ( A ) . Remarks The definition of independence is symmetric with respect to A and B . The product definition applies even if P ( A ) = 0 or P ( B ) = 0. Corollary If A and B are independent, then A and B c are independent. Similarly for A c and B , or for A c and B c . Definition (Conditional independence) We say that A and B are independent conditioned on C , where P ( C ) > 0, if P ( A B C ) = P ( A C ) P ( B C ) . Definition (Independence of a collection of events) We say that events A 1 , A 2 , . . . , A n are independent if for every collection of distinct indices i 1 , i 2 , . . . , i k , we have P ( A i 1 . . . A i k ) = P ( A i 1 ) ⋅ P ( A i 2 )⋯ P ( A i k ) .
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  • Fall '15
  • Probability theory, random variable X

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