Minimizing the probability of a Guassian random variables to be positive

Question

Suppose that $X \sim \mathcal{N}(\mu(M),\sigma^2(M))$, where $\mu \in \mathbb{R} $ and $\sigma^2 \in \mathbb{R}_+$ are functions of its matrix-valued argument $M$. I found in this manuscript that the matrix $M^∗$ which minimizes the probability of $X$ to be positive is given by $$ M^* = \mathrm{argmin}_{M} \frac{\mu(M)}{\sigma^2(M)}$$ May I know how to derive this result?

Answer 1

Uh, it seems you are asking for the value of $M$ to minimize $P(\sigma(M) Z + \mu (M)> 0)$? Well, then rearranging, we get $P(Z > -\mu(M) /\sigma(M)) =: F(-\mu(M)/\sigma(M))$, which is of course a monotone function in its argument, since it is a complementary CDF. So, the result you have (correctly) cited seems to miss a square root?