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Discussion Notes 1

Discussion Notes 1 - EE132B Recitation 1 Probability Review...

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EE132B - Recitation 1 Probability Review
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2 Administrative Stuff TA: Choo Chin (Jeffrey) Tan Email: [email protected] Phone: (310) 384-8078 Office Hours: Friday 11:00-12:00pm (online wimba) Saturday 11:00-12:00pm (by appointment) 9 Recitations & Midterm and Final Rev g g g g g iews Homework Submission: Every Friday via Email g
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3 Outline of Review Probability Axioms Discrete Random Variables Continuous Random Variables Expectation Values and Variances Moment Generating Function g g g g g
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4 Components of a Probability Model ( 29 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 g ( 29 ( 29 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 E S P g g ( 29 us a probability model is a triplet defined as , , . S E P
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5 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 P A g ( 29 ( 29 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. P A P A P A = = g
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6 Terminology and Definitions ( 29 ( 29 ( 29 ( 29 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 A A A A A A = B A B A B = + -
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7 Mutually Exclusive Events ( 29 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 A B A B n = ∅ ( 29 ts 2 are mutually exclusive iff. , , otherwise i i j n A i j A A = =
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8 Probability Axioms ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 For any event , the probability of that evant is such that: 0 1 1 0 1 If events and are not mutually exclusive, ( ) ( ) ( ) ( ) If events , , ..., are all mutua c n A P A P A P S P P A P A P A A B P A B P A P B P A B A A A = ∅ = = = - = + - g g g g ( 29 ( 29 1 1 lly exclusive, then since 0 for . n n i i i i i j P A P A P A A i j = = = = U
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9 Probability Axioms (cont.) ( 29 1 2 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 i i 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 - = = = = = + + + = K g g U 1 1 2 1 where , , .... are disjoint. Sum of total probability : ( ) 1, where 0 ( ) 1 for . k k n i i i B B B P A P A i = = 2200 g
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10 Conditional Probability ( 29 ( 29 ( 29 ( 29 ( 29 The probability of event occurring, given that event has occurred, is called the conditional probability of given .
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