NotesNov1 - X x i E Y | X = x i p X x i if X is discrete...

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Example : Let ( X, Y ) be a continuous random vector with a joint pdf f X,Y ( x, y ) = 15 xy 2 , 0 x, y 1 , y x. Compute the conditional densities f X | Y and f Y | X ; Compute the conditional mean and the con- ditional variance of Y given X = x for 0 < x < 1.
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Computation of expectation and variance via conditioning Using conditional distributions can sometimes simplify computation of means and variances. The idea: the conditional mean E ( Y | X = x ) and the conditional variance Var( Y | X = x ) are functions of the value x of the ran- dom variable X . Since both the conditional mean and the conditional variance are functions of a ran- dom variable X , they can themselves be viewed as random variables.
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Suppose that X and Y are continuous, with a joint pdf f X,Y . EY = Z -∞ Z -∞ y f X,Y ( x, y ) dx dy = Z -∞ E ( Y | X = x ) f X ( x ) dx.
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Abbreviated expression : EY = E ( E ( Y | X )) ; it is true for any random variables ( X, Y ): dis- crete, continuous or mixed. Terminology : the formula of double expecta- tion .
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The exact meaning of the formula of double expectation: EY = E (
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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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