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1. Basic Prob

# 1. Basic Prob - EE 178 Lecture Notes 1 Basic Probability...

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EE 178 Lecture Notes 1 Basic Probability Set Theory Elements of Probability Conditional probability Total Probability and Bayes Rule Independence Counting EE 178: Basic Probability Page 1 – 1

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Set Theory Basics A set is a collection of objects, which are its elements ω A means that ω is an element of the set A A set with no elements is called the empty set , denoted by Types of sets: Finite: A = { ω 1 , ω 2 , . . . , ω n } Countably infinite: A = { ω 1 , ω 2 , . . . } , e.g., the set of integers Uncountable: A set that takes a continuous set of values, e.g., the [0 , 1] interval, the real line, etc. A set can be described by all ω having a certain property, e.g., A = [0 , 1] can be written as A = { ω : 0 ω 1 } A set B A means that every element of B is an element of A A universal set Ω contains all objects of particular interest in a particular context, e.g., sample space for random experiment EE 178: Basic Probability Page 1 – 2
Set Operations Assume a universal set Ω Three basic operations: Complementation: A complement of a set A with respect to Ω is A c = { ω Ω : ω / A } , so Ω c = Intersection: A B = { ω : ω A and ω B } Union: A B = { ω : ω A or ω B } Notation: ◦ ∪ n i =1 A i = A 1 A 2 . . . A n ◦ ∩ n i =1 A i = A 1 A 2 . . . A n A collection of sets A 1 , A 2 , . . . , A n are disjoint or mutually exclusive if A i A j = for all i = j , i.e., no two of them have a common element A collection of sets A 1 , A 2 , . . . , A n partition Ω if they are disjoint and n i =1 A i = Ω EE 178: Basic Probability Page 1 – 3

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Venn Diagrams (e) A B (f) A B (b) A (d) B c (a) Ω (c) B EE 178: Basic Probability Page 1 – 4
Algebra of Sets Basic relations: 1. S Ω = S 2. ( A c ) c = A 3. A A c = 4. Commutative law: A B = B A 5. Associative law: A ( B C ) = ( A B ) C 6. Distributive law: A ( B C ) = ( A B ) ( A C ) 7. DeMorgan’s law: ( A B ) c = A c B c DeMorgan’s law can be generalized to n events: ( n i =1 A i ) c = n i =1 A c i These can all be proven using the definition of set operations or visualized using Venn Diagrams EE 178: Basic Probability Page 1 – 5

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Elements of Probability Probability theory provides the mathematical rules for assigning probabilities to outcomes of random experiments, e.g., coin flips, packet arrivals, noise voltage Basic elements of probability: Sample space : The set of all possible “elementary” or “finest grain” outcomes of the random experiment (also called sample points ) – The sample points are all disjoint – The sample points are collectively exhaustive , i.e., together they make up the entire sample space Events : Subsets of the sample space Probability law : An assignment of probabilities to events in a mathematically consistent way EE 178: Basic Probability Page 1 – 6
Discrete Sample Spaces Sample space is called discrete if it contains a countable number of sample points Examples: Flip a coin once: Ω = { H, T } Flip a coin three times: Ω = { HHH, HHT, HTH, . . .

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1. Basic Prob - EE 178 Lecture Notes 1 Basic Probability...

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