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ECE313.Lecture08 - ECE 313 Probability with Engineering...

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Independent Trials 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 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Review — Fundamentals A random variable X associates a number with each outcome of an experiment Observed value of X varies at random as the experiment is repeated Probabilistic behavior of a discrete random variable is described by its pmf Probabilities of events such as {a < X < b} can be calculated from pmf The pmf of Y = g( X ) can be obtained from the pmf of X
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Review — Expectation The average value of X on repeated trials E[ X ], the expectation of X , is given by E[ X ] = u k •p X (u k ) = µ or µ X where p X (u) denotes the pmf of X Interpretations of E[ X ] Average value of X over many trials Moment about origin of prob. masses µ is the location of the center of mass Fair price to play game with winnings X
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Review — LOTUS The expected value of Y = g( X ) can be found by first finding the pmf of Y from the pmf of X and then using E[ Y ] = v j •p Y (v j )
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 Review — Applications of LOTUS Y = a X + b, where a and b are constants. E[ Y ] = a•E[ X ] + b µ Y = a•µ X + b E[a X + b] = a•E[ X ] + b Expectation is a linear operation: the expectation of a sum is the sum of the expectations E[a X ] = a•E[ X ] E[b] = b Masses in the pmf of Y = a X are “further away” from the origin by a factor of a, and so is the center of mass “further away”
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 This will hurt for just a few moments … E[ X – a] is the (first) moment of X about a E[( X – a) n ] = n-th moment of X about a E[ X n ] is called the n-th moment of X E[( X – µ) n ] is the n-th central moment of X E[ X – µ], the first central moment of X is 0 E[( X – µ) 2 ], the second central moment of X , is commonly called the variance of X
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 Review — Variance I E[( X – µ) 2 ], the second central moment of X , is also known as the variance of X Variance is denoted by var( X ) or σ 2 or σ X 2 where σ is called the standard deviation For a discrete random variable, LOTUS says σ 2 = E[( X – µ) 2 ] = (u k – µ) 2 •p X (u k ) σ > 0 except for trivial random variables
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ECE 313 - Lecture 8 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 40
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