Name of the following minimization $E[(X - c)^2] = Var(X) + (E[X] - c)^2$ with $c = E[X]$

My professor proposed the below relationship as a property of the variance (he called $$E[(X - c)^2]$$ mean squared error):

$$E[(X - c)^2] = Var(X) + (E[X] - c)^2$$

and he said that, when $$c = E[X]$$, the mean squared error is minimized.

I'm wondering which is the name of this property/relationship.

• This is the statistics version of Pythagoras' Theorem. Commented Feb 17, 2022 at 17:01

The identity you have is simply the bias-variance decomposition of the mean squared error: $$E[(X - c)^2] = Var(X) + (E[X] - c)^2\quad (1),$$
and it is proven quite simply by rearranging the identity for the variance $$Var(Y)=E[Y^2]-(E[Y])^2$$
applied to the case $$Y=X-c$$.
Take $$X$$ as given and $$c$$ as a choice argument. Then it is clear from the right hand side of $$(1)$$ that since the variance term is independent of $$c$$ and the second term (bias squared) is nonnegative, the whole expression is minimized if you choose $$c=E[X]$$ (the bias term is then zero).