# MSE Minimized by Mean

In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $$X \in L^2(\mathcal{F};\mathbb{P})$$, is a random-variable in $$\mathbb{R}$$, then is the does $$\mathbb{E}_{\mathbb{P}}[X] = \inf_{Z \in L^2(\mathcal{F};\mathbb{P})} \;\mathbb{E}_{\mathbb{P}}\left[(X-Z)^2\right]?$$

The argmin, is the conditional expectation, but what about the min itself?

• ..... variance? – pointguard0 Feb 5 at 15:29
• Probably not, since that isn't true for a 1-dimensional Gaussian. – N00ber Feb 5 at 15:36