I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no concrete findings. I want to pose this questions here and take some feedback on how we can solve it. So lets assume you have a Gaussian distribution $X\sim \mathcal{N}(\mu, \sigma^2)$, with $\mu\geq 0$. Let $$\bar{\mu} = \frac{1}{n} \sum_{i=1}^n x_i$$ be the sample mean of $X$, and let $$\bar{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{\mu})^2$$ be its biased estimator (you can assume unbiased estimator too). So now I want to find $$\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right],$$ i.e. the expectation of sample mean to sample variance. If this is not achievable then then a concentration inequality on $$\mathbb{P} \left(\frac{\bar{\mu}}{\bar{\sigma}^2} - \frac{\mu}{\sigma^2} >\epsilon\right)$$ My solutions have not found anything solid and I don't think my ways are viable. Anyone who can give any hints on how to solve it would be appreciated.
PS: One user suggested to see t-distribution, but to use it we need the knowledge of mean.
