Let $X_1,\dots,X_n$ be Gaussian random variables with mean $\mu$ and variance $\sigma^2$. What is known about the distribution of
$\frac{\sqrt{\frac1{n-1}\sum_{i=1}^n(X_i-\bar X)^2}}{\bar X} $
with $\bar X = \frac 1n \sum_{i=1}^n X_i$?
Does this distribution have a name? I am especially interested in its cumulative distribution function.
If this is too complicated, I'd settle for
$\frac{\sqrt{\frac1{n}\sum_{i=1}^n(X_i-\mu)^2}}{\mu} $