lecture02

# Lecture02 - ECE 5510 Random Processes Lecture Notes Fall 2009 Lecture 2 Today(1 Events as Sets(2 Axioms and Properties of Prob ability(3

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 12:30-2pm, 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 (6-sided) die. • S = { Adenine,Cytosine,Guanine,Thymine } for the nucleotide found at a particular place in a strand of DNA. • S = C , i.e. , non-negative 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...
View Full Document

## 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.

### Page1 / 7

Lecture02 - ECE 5510 Random Processes Lecture Notes Fall 2009 Lecture 2 Today(1 Events as Sets(2 Axioms and Properties of Prob ability(3

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online