| (x | = conditional density of given (Y = fX Y y) X, y) = E( | = ) = = X Y y This is a function of the given value Since we can think of as a y. y carrier (or dummy) symbol for the expression E( | ) should be Y, X Y regarded as a random variable which is a function of Y. Since E( | ) is a random variable which is a function of we can get its X Y Y, expectation by integrating relative to the density of This work gives Y. E { E( | ) } = X Y = = The previous step reversed the order of integration. In problems in which and are distributed over a X Y twodimensional set which is not a rectangle, this can be very tricky. The inner integral in { } is fX(x). = = E( ) X This is a very important finding. This works as well for any function of say X, as is just another random variable. Thus it happens that g(X), g(X) E { E( | ) } = E( ). g(X) Y g(X) Var( | = ) = E[ | = ] { E( | = ) }2 X Y y X2 Y y X Y y This is a function of the given value Since we can think of as a y. y carrier (or dummy) symbol for the expression Var( | ) should be Y, X Y regarded as a random variable which is a function of Y. Examine now E{ Var (X | ) }. Of course, E(X | ) is some function of and it Y Y Y, may help to give it a temporary name. Let it be h(Y). E{ Var (X | ) } = E{ E[ | ] ( E( | ) ) 2 } Y X2 Y X Y
fX | Y (x | y) = conditional density of X, given (Y = y) = E( X | Y = y ) = = This is a function of the given value y. Since we can think of y as a carrier (or dummy) symbol for Y, the expression E( X | Y ) should be regarded as a random variable which is a function of Y. Since E( X | Y ) is a random variable which is a function of Y, we can get its expectation by integrating relative to the density of Y. This work gives E { E( X | Y ) } = = = The previous step reversed the order of integration. In problems in which X and Y are distributed over a twodimensional set which is not a rectangle, this can be very tricky. The inner integral in { } is fX(x). = = E( X ) This is a very important finding. This works as well for any function of X, say g(X), as g(X) is just another random variable. Thus it happens that E { E( g(X) | Y ) } = E( g(X) ). Var( X | Y = y ) = E[ X2 | Y = y ] { E( X | Y = y ) }2 This is a function of the given value y. Since we can think of y as a carrier (or dummy) symbol for Y, the expression Var( X | Y ) should be regarded as a random variable which is a function of Y. Examine now E{ Var (X | Y ) }. Of course, E(X | Y ) is some function of Y, and it may help to give it a temporary name. Let it be h(Y). E{ Var (X | Y ) } = E{ E[ X2 | Y ] ( E( X | Y ) ) 2 }
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