Statistik 3 - Beispiel 2

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Zumindest (a)

\operatorname{E}(X) = \mu_X

\operatorname{E}(Y) = \mu_Y


 \sigma_{X,Y} = 
 \operatorname{E}( X ( Y - \mu_Y)) = 
 \operatorname{E}(XY - X \mu_Y) =
 \operatorname{E}(EY) - \mu_Y \operatorname{E}(X) =
 \operatorname{E}(EY) - \mu_Y \mu_X


 \sigma_{X,Y} = 
 \operatorname{E}( ( X - \mu_X) Y) = 
 \operatorname{E}(XY - \mu_X Y) =
 \operatorname{E}(EY) - \mu_X \operatorname{E}(Y) =
 \operatorname{E}(EY) - \mu_X \mu_Y


 \sigma_{X,Y} = 
 \operatorname{E}( ( X - \mu_X) ( Y - \mu_Y) ) =


 \operatorname{E}(XY - \mu_X Y - X \mu_Y + \mu_X \mu_Y) =


 \operatorname{E}(EY) - \mu_X \operatorname{E}(Y) - \mu_Y \operatorname{E}(X) + \mu_X \mu_Y =


 \operatorname{E}(EY) - \mu_X \mu_Y - \mu_Y \mu_X + \mu_X \mu_Y =


 \operatorname{E}(EY) - \mu_X \mu_Y