Lec. 3 - Sept 4, 2019 - Probability Axioms, Their Applications.pdf - Axiomatic Definition of Probability and its Application Dr Madhu S Gupta RF

Lec. 3 - Sept 4, 2019 - Probability Axioms, Their Applications.pdf

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9/6/2019 S AN D IEGO S TATE U NIVERSITY 1 Axiomatic Definition of Probability, and its Application Dr. Madhu S. Gupta RF Communication Systems Industry Chair Professor Department of Electrical and Computer Engineering & Director Communication Systems & Signal Processing Institute San Diego State University San Diego, California 92182
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 2 Outline 1. Constructing complex events by logical operators 2. Venn Diagrams for visualizing the events 3. Meaning of “Axiomatic” 4. Three basic Axioms 5. Consequences (“Propositions”) following from the Axioms 6. Addition Rule: Probability of the union of two events 7. Special case for Mutually-exclusive events 8. Calculating event probabilities from axioms and propositions 1. Union of disjoint events 2. Union of non-disjoint events 3. Intersection of independent events 4. Complement of an event 5. Partitioning of sample space
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 3 Event Probability Definitions Encountered Relative- Frequency Definition N TR = number of trials conducted n A = number of occurrences of the event A -- Requires experimental determination Definition Based on Equally-Likely Set of Outcomes N EL = number of equally-likely outcomes in the sample space n A = number of outcomes leading to the event A -- Requires an equally-likely set of outcomes Not Sufficient in practice, because Large number of trials not practical. Outcomes usually not equally-likely in practice. Events of interest in practice are compound (complex combinations of elementary) events. Search for general rules of probability assignment [ ] TR A N TR n P A N Lim  [ ] A EL n P A N
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 4 Defining Compound Events
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 5 Construction of More Complex Events (1) Union of Two Events (2) Intersection of Two Events (3) Complement of an Event A A B ' A A B B
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 6 Construction of Compound Events Joint Occurrence of Two Events By definition, and event = a collection of outcomes. Union of Events, A B . A compound event that occurs whenever either event A, or event B, or both occur. Intersection of Events, A B “(Joint Occurrence”). The set of outcomes common between events A and B . If the events A and B do have some outcomes in common, then the occurrence of those outcomes implies that both events A and B have occurred simultaneously (“joint” occurrence). Complement of an Event, A’. The set of all those outcomes in S that are not in A.
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9/6/2019 S AN D IEGO S TATE U NIVERSITY 7 Disjoint (or “Mutually-Exclusive”) Events Definition: If the two events A and B are non-overlapping, i.e., they do not have any common outcomes: Then A and B are “disjoint” or mutually exclusive.
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