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**Unformatted text preview: **X x i E ( Y | X = x i ) p X ( x i ) if X is discrete with pmf p X ; EY = E ( E ( Y | X )) = Z ∞-∞ E ( Y | X = x ) f X ( x ) dx, if X is continuous with pdf f X . Example : The number of claims arriving to an insurance company in a week is a Pois-son random variable N with mean 20. Assume that the amounts of diﬀerent claims are inde-pendent exponentially distributed random vari-ables with mean 800. Assume that the claim amounts are indepen-dent of the number of claims arriving in a week. Find the expected total amount of claims the company receives in a week. • The formula of the double expectation EY = E ( E ( Y | X )) is a device to compute expectations by con-ditioning . • There is a device to compute variances by conditioning : for any two random variables X and Y Var Y = E (Var( Y | X )) + Var( E ( Y | X )) ....

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- Fall '09
- SAMORODNITSKY
- Variance, Probability theory, conditional variance, joint pdf fX,Y