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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 2 Today: (1) Events as Sets (2) Axioms and Properties of Prob ability (3) Independence of Sets Announcements: • HW 1 assigned today, due a week from today, 5pm. • Please sign up for a date for your discussion item. • Office Hours today from 12:302pm, because of a disserta tion defense (2:15, Adam Wunderlich, INC Main Conference Room). 1 Events as Sets All probability is defined on sets. In probability, we call these sets events . A set is a collection of elements. In probability, we call these outcomes . Def’n: Event A collection of outcomes. Order doesn’t matter, and there are no duplicates. 1.0.1 Set Terminology vs. Probability Terminology Set Theory Probability Theory Probability Symbol universe sample space (certain event) S element outcome (sample point) s set event E disjoint sets disjoint events E 1 ∩ E 2 = ∅ null set null event ∅ 1.1 Introduction There are different ways to define an event (set): • List them: A = { , 5 , 10 , 15 ,... } ; B = { Tails,Heads } • As an interval: [0 , 1], [0 , 1), (0 , 1), ( a,b ]. Note overlap with coordinates! ECE 5510 Fall 2009 2 • An existing event set name: N , R 2 , R n • By rule: C = { x ∈ R : x ≥ } , D = { ( x,y ) ∈ R 2 : x 2 + y 2 < R 2 } . Note Y&G uses ‘  ’ instead of the colon ‘:’, which I find confusing. 1.1.1 Important Events Here’s an important event: ∅ = {} , the null event or the empty set . Here’s the opposite: S is used to represent the set of everything possible in a given context, the sample space . • S = B above for the flip of a coin. • S = { 1 , 2 , 3 , 4 , 5 , 6 } for the roll of a (6sided) die. • S = { Adenine,Cytosine,Guanine,Thymine } for the nucleotide found at a particular place in a strand of DNA. • S = C , i.e. , nonnegative real numbers, for your driving speed (maybe when the cop pulls you over). 1.2 Finite, Countable, and Uncountable Event Sets We denote the size of, i.e. , the number of items in, a set A as  A  . If  A  is less than infinity then set A is said to be finite . But there are two kinds of infinite sets: Countably Infinite: The set can be listed. That is, each element could be assigned a unique positive integer. Eg. { 1 , 2 , 3 ,... } , or...
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This note was uploaded on 09/15/2011 for the course ECE 5510 taught by Professor Chen,r during the Fall '08 term at University of Utah.
 Fall '08
 Chen,R

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