Minimise the probability of a variable being positive?

Question

So the problem statement is that we are given that ($v^Ts$ is the variance) $$x \sim \mathcal{N}(v^Tc,v^Ts)$$ Where $c,s$ are constants and $v^Tv=1$. Show that minimising $\mathbb{P}(x>0)$ with respect to $v$ is equivalent to minimising $\frac{v^Tc}{\sqrt{v^Ts}}$.

Answer 1

Hint: In order to minimize $P(x > 0)$ you want the mean of your Gaussian to be as many standard deviations above zero as possible. That's exactly the interpretation of the second quantity you mentioned in terms of $v,c,s$.