LEC08 Mean & Variance

LEC08 Mean & Variance - ( 29 [ ] V X E X ( ) =- 2 SD X...

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Expected Value of a Discrete Random Variable discrete r.v. X possible values x 1 , x 2 , . .. pdf f(x)=P(X=x) E X x f x x P X x i i i i i i ( ) ( ) ( ) = = = Called: Expected Value of X Mean of X E(X) μ
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What is difference between "average" and mean? Mean ==== Population mean is a population parameter mean is a constant Average === Sample average is computed from a sample average is a random variable average will be different for each new sample 2
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Example The number of defects in a manufactured unit has the following pdf: r.v. X = number of defects x 0 1 2 f(x) .3 .5 .2 E(X) = 0 (.3) + 1 (.5) + 2 (.2) = .9 E(X) does not equal one of the possible values E(X) vs. sample average For n=100 assemblies, the sample average will not be 0.9 exactly For n →∞ assemblies, the population "average" equals E(X)= 0.9 3
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Variance of a Discrete r.v. X
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Unformatted text preview: ( 29 [ ] V X E X ( ) =- 2 SD X V X ( ) ( ) = Unit of V(X) is defects 2 , inches 2 or grams 2 etc. Unit of SD(X) is defects, inches or grams; same as for X Other names V(X)=Var(X)= 2 (sigma-squared) 2 is always positive or zero 4 Compute V(X) r.v. X = number of defects x 1 2 f(x) .3 .5 .2 ( 29 X- 2 .81= (0-.9) 2 .01= (1-.9) 2 1.21= (2-.9) 2 E(X) = 0 (.3) + 1 (.5) + 2 (.2) = .9 ( 29 [ ] V X E X ( ) =- 2 = .81(.3) + .01 (.5) + 1.21 (.2) = 0.49 5 Compute V(X) a Better way r.v. X = number of defects x 1 2 f(x) .3 .5 .2 X 2 1 4 Use this algebraic equivalence: V X E X ( ) ( ) =-2 2 =0.9 and 2 = .81 E(X) = 0(.3) + 1(.5) + 4(.2) = 1.3 V(X) = 1.3 - (.9) 2 =.49 SAME! 6...
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LEC08 Mean & Variance - ( 29 [ ] V X E X ( ) =- 2 SD X...

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