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lecture02 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 2 Today: (1) Events as Sets (2) Axioms and Properties of Prob- ability (3) Experimental RSS Measurement Announcements: HW 1 assigned today, due a week from today. Please sign up for a date for your discussion item. The application assignment 1 is posted on the wiki / WebCT. For quicker browsing for non-WebCT items, my wiki is span. ece.utah.edu and then go to Teaching: ECE 5510. You can also get there from WebCT. 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 = { 0 , 5 , 10 , 15 ,... } ; B = { Tails,Heads } As an interval: [0 , 1], [0 , 1), (0 , 1), ( a,b ]. Note overlap with coordinates!
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ECE 5510 Fall 2008 2 An existing event set name: N , R 2 , R n By rule: C = { x R : x 0 } , 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).
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