Lecture 4 - Concepts of Expected Value and Expectation...

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Concepts of Expected Value and Expectation Illustration of the mean value calculation Illustration of the variance calculation Variance shortcut formula V(X) = E(X 2 ) – [E(X)] 2
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• Estimator – Formula or rule for estimating a parameter value from sample data. to estimate μ • An estimator is unbiased if its expected value equals the parameter value it estimates. X
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x x 2 p(x) 1 E(x) x p(x) V(x) E((x E(x)) μ = = ⋅= =− 2 22 (( ) ) ( ) (2 ( ) [ ( ) ] ) ( ) x x xE x p x x x E x p x +
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2 2 22 () 2 [ ] [( ) ] ( ) 2 ( ) ( ) [ ( ) ] ( ) [ ] xx x x Vx x p x x p x E x p x Ex px Ex Ex Vx Ex =⋅ +⋅ =− + ∑∑ Second Moment or “Squared Coefficient of Variation”
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/ 1 () ( ) 1 i i i i xx n Ex n nx n μμ = =⋅ = =∴ Is an unbiased Estimator of u
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2 2 () 1 i i xx s n = Is s 2 an unbiased estimator for σ 2 ? Yes! 22 2 2 2 / i Ex n μσ ⎛⎞ =+ ⎜⎟ ⎝⎠
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Problem 3-29 (page 106) Illustration of application of the pmf to obtain parameter values Problem 3-30 (page 106) Illustration of expectation
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Problem 3-35 (page 107) Application of expected value calculations, probability modeling
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3-29 • Data Rich Information Poor (DRIP) x 0 1 2 3 4 p(x) .01 .15 .45 .27 .05 Raw Data Histogram Parameters E(x)=0(.08)+1(.15)+…4(.05)=2.06
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A probability distribution tells us much more that just “μ” 0 Loss or risk profit
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This note was uploaded on 04/08/2008 for the course ENGR 2600 taught by Professor Malmborg during the Spring '08 term at Rensselaer Polytechnic Institute.

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Lecture 4 - Concepts of Expected Value and Expectation...

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