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NotesNov3

# NotesNov3 - if X is continuous with pdf f X Var E Y | X = X...

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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.

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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 )) .
The exact meaning of terms in the formula for the variance: E (Var( Y | X )) = X x i Var( Y | X = x i ) p X ( x i ) if X is discrete with pmf p X ; E (Var( Y | X )) = Z -∞ Var( Y | X = x ) f

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Unformatted text preview: if X is continuous with pdf f X . Var( E ( Y | X )) = X x i ( E ( Y | X = x i )) 2 p X ( x i )- X x i E ( Y | X = x i ) p X ( x i ) 2 if X is discrete with pmf p X ; Var( E ( Y | X )) = Z ∞-∞ ( E ( Y | X = x )) 2 f X ( x ) dx-±Z ∞-∞ E ( Y | X = x ) f X ( x ) dx ² 2 if X is continuous with pdf f X . Example : Back to the example where the number of claims arriving to an insurance com-pany in a week is a Poisson random variable N with mean 20, and the amounts of diﬀer-ent claims are independent exponentially dis-tributed random variables with mean 800. Compute the variance of the total claim amount the company receives in a week....
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