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EE132B - Recitation 1 Probability Review

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2 Outline of Review Probability Axioms Discrete Random Variables Continuous Random Variables Expectation Values and Variances Moment Generating Function
3 Components of a Probability Model   An experiment is the process of observing a phenomenon with multiple possible outcomes. Sample Space : A set of all possible observable outcomes of a random phenomena. The sample space may be discrete S     or continuous. Set of Events : A set (collection) of one or more outcomes in the sample space, where . Probability of Events : A consistent description of the likelihood of observing an event. Th E ES P   us a probability model is a triplet defined as , . S E P

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4 Probability Probability of an event estimates the proportion of times the event is expected to occur in repeated random experiments, and is denoted as ( ). Some properties: Probability values are always between A PA     0 ( ) 1 . Probability is a numerical value of the likelihood that an event will occur. 0 indicates an event that is never/impossible to occur. 1 indicates an event that is certain to occur. 
5 Terminology and Definitions         Given events , , Union of two events: or Intersection of two events: and Complement of an event: Not Cardinality (Size) of Sets: Let the number of elements of the set size of A B S A B A B A B A B AA A A A A B A B A B

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6 Mutually Exclusive Events   Sample space is a set and events are the subsets of this (universal) set. Two events and are mutually exclusive (disjoint) iff. if and only if their intersection is empty, i.e. A set of even AB n     ts 2 are mutually exclusive iff. , otherwise i ij n A i j AA 
7 Probability Axioms               12 For any event , the probability of that evant is such that: 01 1 0 1 If events and are not mutually exclusive, ( ) ( ) ( ) ( ) If events , , . .., are all mutua c n A P A PA PS P P A P A P A AB P A B P A P B P A B A A A         1 1 lly exclusive, then since 0 for . n n ii i i ij P A P A P A A i j   

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8 Probability Axioms (cont.)   12 1 1 2 1 2 1 1 Given events { , , , }, and the probability of each outcome ( ), Sum of disjoint products : ... ... Total probability theorem : ( | ) ( ) n i i n i n n i ii i E A A A p P e P A P(A ) P( A A ) P( A A A A ) P A P A B P B     1 1 where , , . ... are disjoint. Sum of total probability : ( ) 1, where 0 ( ) 1 for . k k n i i i B B B PA P A i
9 Conditional Probability           The probability of event occurring, given that event has occurred, is called the conditional probability of given .

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