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You are here: Home Stats Department Introduction to probability - IN TRANSLATION Samdreifingar Samdreifni óháðra hendinga

Samdreifni óháðra hendinga

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Setning: Ef $ X$ og $ Y$ eru óháðar hendingar, þá er $ Cov(X,Y)=0$ .






Sönnun setningarinnar: f sé samþéttifall $ X$ og $ Y$ .


$\displaystyle Cov(X,Y)$ $\displaystyle =$ $\displaystyle E[(X-\mu_X)(Y-\mu_Y)]$  
  $\displaystyle =$ $\displaystyle E[(X-\mu_X)Y] - E[(X-\mu_X)\mu_Y]$  
  $\displaystyle =$ $\displaystyle E[XY] -E[\mu_XY] - E[(X-\mu_X)]\mu_Y$  
  $\displaystyle =$ $\displaystyle E[X]E[Y] -\mu_X E[Y] - (E[X]-\mu_X)\mu_Y$  
  $\displaystyle =$ 0  

en í síðustu línunni er notað að $ E[X]=\mu_X$ og tilsvarandi fyrir $ Y$ .

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