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You are here: Home Stats Department Introduction to probability - IN TRANSLATION Samdreifingar Meira um væntigildi hendinga

Meira um væntigildi hendinga

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Setning: Ef $ X$ og $ Y$ eru hendingar og væntigildi þeirra er til, þá er $ E[X+Y]=E[X]+E[Y]$ .




Sönnun: ... fyrir strjálar hendingar.


$\displaystyle E[X+Y]$ $\displaystyle =$ $\displaystyle \sum_x \sum_y \left ( x + y \right ) p(x,y)$  
  $\displaystyle =$ $\displaystyle \sum_x \sum_y x p(x,y) \quad + \quad \sum_x \sum_y y p(x,y)$  
  $\displaystyle =$ $\displaystyle \sum_x \left \{ \sum_y x p(x,y) \right \} \quad + \quad \sum_y \left \{ \sum_x y p(x,y) \right \}$  
  $\displaystyle =$ $\displaystyle \sum_x x \left \{ \sum_y p(x,y) \right \} \quad + \quad \sum_y y \left \{ \sum_x p(x,y) \right \}$  
  $\displaystyle =$ $\displaystyle \sum_x x p_X(x) \quad + \quad \sum_y y p_Y(y)$  



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