Minimizer of Perturbed Mean

Question

Suppose that $X \in L^2(\Omega,\mathcal{F},\mathbb{P})$, where $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space. I know that the minimizer of $$ \inf_{r \in \mathbb{R}} \mathbb{E}\left[ (r-X)^2 \right], $$ is the mean of $X$. My question, is how could we solve for the minimizer of $$ \inf_{r \in \mathbb{R}} \mathbb{E}\left[ (r-X)^2 \right]-r? $$

Answer 1

Begin with the ansatz that $\mathbb{E}[(r-X)^2]$ is differentiable as a function of $r$ (or show it), then computing the gradient and setting it to $0$, we see that $$ \begin{aligned} 0=&\nabla \mathbb{E}[(r-X)^2] - r = \mathbb{E}[2(r-X)]-1\\ r & = \mathbb{E}[X]+\frac1{2} = \mu_X + \frac1{2}. \end{aligned} $$

Just make sure that the ansatz works in general; I'll leave that part to you :)