of another random variable (X) that we can use to predict Y, we can potentially use this knowledge to reduce the expected squared error. As it turns out, the best prediction of Y given X is the conditional expectation. In particular, for any measurable,

By selecting , the second, nonnegative term becomes zero, showing the claim.Here, the second equality used the law of total expectation.We also see that the expected conditional variance of Y given X shows up as the irreducible