ECE313.Lecture06

ECE313.Lecture06 - ECE 313 Probability with Engineering...

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Random Variables Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 Summary of previous lectures Probability and statistics in general terms Subjective, classical, and relative frequency approaches to probability Probability is a numerical quantiFcation of our beliefs about future occurrences An event of probability p is expected to occur roughly pN times on N trials Axiomatic approach to probability
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 Show me where it is written … Where in the axiomatic theory does it say that an event of probability p is expected to occur pN times on N future trials? To understand the where, we need to develop two important concepts: random variables independent trials Both concepts have been known, and even well-understood, since ancient times
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 It’s not whether you won or lost, … Consider an experiment of rolling a fair die Suppose that you bet one dollar on the outcome of the trial If the outcome is a perfect square {1, 4} you win one dollar — of course, you do get your original dollar back too! Otherwise, if the outcome is {2, 3, 5, 6}, you lose the dollar that you bet Your wealth increases/decreases by $1
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 … it’s how you played the game This game can be played in different ways Scenario #1: You give your bookie $1 as a bet. If your winning number comes up, he returns $2 to you — else he waves to you while he laughs all the way to the bank Scenario #2: You tell your bookie you want to bet $1. If your winning number comes up, he sends you $1 — else he sends two large men with baseball bats to collect $1
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 In the end, it is all the same Regardless of which scenario is your bookie’s modus operandi, it is always true that at the end of the game, your net worth has either increased or decreased by $1 An alternative viewpoint of Scenario #1 is that your $1 bet is a nonreturnable entry fee to play the game, and you win $2 or $0 Scenario #1 is the one commonly used in lotteries, bets on horse races, and the like
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ECE 313 - Lecture 6 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 36 You win some, you lose some … Let X denote your “winnings” in this game X takes on values +1 or –1 depending on the outcome of the trial Positive values of X
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This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture06 - ECE 313 Probability with Engineering...

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