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

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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
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Axioms of Probability A probability measure P satisfies the following axioms: 1. P( A ) 0 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 = j — then P i =1 A i = 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 su ffi cient 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, TH, TTH, TTTH, . . . } Number of packets arriving at a node in a communication network in time interval (0 , T ] : Ω = { 0 , 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
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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 { ω } — so that: P( {
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