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## Homework Statement

Let X and Y be two unknown variables with E(Y)=[tex]\mu[/tex] and EY

^{2}< [tex]\infty[/tex].

## Homework Equations

a. Show that the constant c that minimizes E(Y-c)

^{2}is c=[tex]\mu[/tex].

b. Deduce that the random variable f(X) that minimizes

*E*[(Y-f(X))

^{2}|X] is f(X)= E[Y|X].

c. Deduce that the random variable f(X) that minimizes

*E*[(Y-f(X))

^{2}] is also f(X)= E[Y|X].

## The Attempt at a Solution

a. E(Y-c)

^{2}= E[Y

^{2}-2Y*c+c

^{2}]

other ideas: Can I do the derivative w.r.t. Y, set it equal to zero. But there is an expectation operator, how do you take expectation through the operator.

I can't use a posterior mean, the problem did not specify distr.of the r.v.

b, c. no ideas yet

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