# Probability theory: minimizing expectation

Let X and Y be two random variables with E(Y)=u and EY^2 < infinity.

a) show that the constant c that minimizes E(Y-c)^2 is c=u

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]

https://brainmass.com/statistics/random-variables/probability-theory-minimizing-expectation-372794

#### Solution Summary

This solution helps explore how to minimize expectation within the context of probability theory.

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