Transcription of Expected Value, Variance and Covariance
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Expected ValueVarianceCovarianceExpected Value, Variance and Covariance (Sections )1 STA 256: Fall 20191 This slide show is an open-source document. See last slide for / 31 Expected ValueVarianceCovarianceOverview1 Expected Value2 Variance3 Covariance2 / 31 Expected ValueVarianceCovarianceDefinition for Discrete Random VariablesThe Expected value of a discrete random variable isE(X) = xxpX(x)Provided x|x|pX(x)< .If the sum diverges, theexpected value does not is only an issue for infinite sums (and integralsover infinite intervals).3 / 31 Expected ValueVarianceCovarianceExpected value is an averageImagine a very large jar full of is the balls are numberedx1,.., aremeasurements carried out on members of the for now that all the numbers are ball is selected at random; all balls are equally likely tobe the number on the ball (X=xi) = (X) = xxpX(x)=N i=1xi1N= Ni=1xiN4 / 31 Expected ValueVarianceCovarianceFor the jar full of numbered balls,E(X) = Ni=1xiNThis is the common average, or arithmetic there are values arevi, fori= 1.
Expected ValueVarianceCovariance Conditional Expectation The idea Consider jointly distributed random variables Xand Y. For each possible value of X, there is a conditional distribution of Y. Each conditional distribution has an expected value (sub-population mean). If you could estimate E(YjX= x), it would be a good way to predict Y from X.
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