Let $X_i \sim N(\mu,\sigma^2)$ where both $\mu,\sigma$ are unknown. Is it possible to find the distribution of $\frac{X_1-\bar{X}}{s^2}$ , where $s^2$ is the sample variance.
It seems difficult to guess the nature of the distribution as $X_1-\bar{X} $ has a normal distribution while $s^2$ has chi-squared distribution and both are not independent. But is there a way to compute the distribution?