Expected Value, Variance and Covariance

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Expected ValueVarianceCovarianceExpected Value, Variance and Covariance (Sections )1STA 256: Fall 20191This slide show is an open-source document. See last slide for / 31Expected ValueVarianceCovarianceOverview1Expected Value2Variance3Covariance2 / 31Expected 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 / 31Expected 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 / 31Expected ValueVarianceCovarianceFor the jar full of numbered balls,E(X) = Ni=1xiNThis is the common average, or arithmetic there are values arevi, fori= 1.

De nition of Covariance Let Xand Y be jointly distributed random variables with E(X) = xand E(Y) = y. The covariance between Xand Y is Cov(X;Y) = E[(X X)(Y Y)] If values of Xthat are above average tend to go with values of Y that are above average (and below average Xtends to go with below average Y), the covariance will be positive.

  Between, Covariance, Covariance between