Transcription of SomeFormulasofMeanandVariance: Weconsidertwo ...
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SomeFormulasofMeanandVariance: Weconsidertwo ...
Some Formulas of Mean and Variance:We consider tworandom :E(X+Y)=E(X)+E(Y).Proof:For discrete random variablesXandY, it is given by:E(X+Y)= i j(xi+yj)fxy(xi,yj)= i jxifxy(xi,yj)+ i jyjfxy(xi,yj)=E(X)+E(Y).119 For continuous random variablesXandY, we can show:E(X+Y)= (x+y)fxy(x,y)dxdy= xfxy(x,y)dxdy+ yfxy(x,y)dxdy=E(X)+E(Y). :E(XY)=E(X)E(Y), whenXis indepen-dent :For discrete random variablesXandY,E(XY)= i jxiyjfxy(xi,yj)= i jxiyjfx(xi)fy(yj)=( ixifx(xi))( jyjfy(yj))=E(X)E(Y).IfXis independent ofY, the second equality holds, ,fxy(xi,yj)=fx(xi)fy(yj).
E(aX) =aE(X) in the second equality and E(X +Y) = E( X )+E( Y ) in the third equality are utilized, where X and Y are random variables and a is a constant value.
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