9.3 In the approximation of the random variable $Y$ by $? (X)$ , one may use the mean cost $^{?} E {g [Y ? ? (X)]}$ , where $g (x)$ is a given function. Show that, if $g (x)$ is an even concave function as below, then the "mean cost" is minimum if $? (X) = E {Y ? x}$ . You may assume the conditional density of $Y$ given the event ${X = x}$ is symmetric abont its mean, as shown below.

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9.3 In the approximation of the random variable

Y

by

? (X)

, one may use the mean cost

^{?} E {g [Y ? ? (X)]}

, where

g (x)

is a given function. Show that, if

g (x)

is an even concave function as below, then the "mean cost" is minimum if

? (X) = E {Y ? x}

. You may assume the conditional density of

Y

given the event

{X = x}

is symmetric abont its mean, as shown below.

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