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# lect01 - Lecture Notes 1 Review of Basic Probability Theory...

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Unformatted text preview: Lecture Notes 1 Review of Basic Probability Theory • Probability Space and Axioms • Basic Laws • Conditional Probability and Bayes Rule • Independence EE 278: Basic Probability 1 – 1 Probability Theory • 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 theory: ◦ Sample space Ω : set of all possible “elementary” or “finest grain” outcomes of the random experiment ◦ Set of events F : set of (all?) subsets of Ω — an event A ⊂ Ω occurs if the outcome ω ∈ A ◦ Probability measure P : function over F that assigns probabilities to events according to the axioms of probability (see below) • Formally, a probability space is the triple (Ω , F , P) EE 278: Basic Probability 1 – 2 Axioms of Probability • A probability measure P satisfies the following axioms: 1. P( A ) ≥ for every event A in F 2. P(Ω) = 1 3. If A 1 , A 2 , . . . are disjoint events — i.e., A i ∩ A j = ∅ , for all i 6 = j — then P ∞ [ i =1 A i = ∞ X i =1 P( A i ) • Notes: ◦ P is a measure in the same sense as mass , length , area , and volume — all satisfy axioms 1 and 3 ◦ Unlike these other measures, P is bounded by 1 (axiom 2) ◦ This analogy provides some intuition but is not sufficient to fully understand probability theory — other aspects such as conditioning and independence are unique to probability EE 278: Basic Probability 1 – 3 Discrete Probability Spaces • A sample space Ω is said to be discrete if it is countable • Examples: ◦ Flipping a coin: Ω = { H, T } ◦ Rolling a die: Ω = { 1 , 2 , 3 , 4 , 5 , 6 } ◦ Flipping a coin n times: Ω = { H, T } n , sequences of heads/tails of length n ◦ Flipping a coin until the first heads occurs: Ω = { H, T H, T T H, T T T H, . . . } ◦ Number of packets arriving at a node in a communication network in time interval (0 , T ] : Ω = { , 1 , 2 , 3 , . . . } The first three examples have finite Ω , whereas the last two have countably infinite Ω . Both types are considered discrete • For discrete sample spaces, the set of events F can be taken to be the set of all subsets of Ω , sometimes called the power set of Ω EE 278: Basic Probability 1 – 4 • Example: For the coin flipping experiment, F = {∅ , { H } , { T } , Ω } • F does not have to be the entire power set (more on this later) • The probability measure P can be defined by assigning probabilities to individual outcomes — single outcome events {...
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lect01 - Lecture Notes 1 Review of Basic Probability Theory...

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