Let $X_1, \dots, X_n$ be independent normally distributed random variables. What is the distribution of:
$$
Y_i = \frac{X_i}{\mathrm{stdDev}(X_1, \dots, X_n)},
$$
where $\mathrm{stdDev}(X_1, \dots, X_n)$ is the standard deviation of the sample? I came across this in a simulation, where the simulated random variables were "normalised" before being used, but no statistical analysis was provided.