Let $X_1,...,X_m$ and $Y_1,...,Y_n$ be independent random samples from normal distributions $N(\mu_1,\sigma^2)$, $N(\mu_2,\sigma^2)$ respectively, where all parameters are unknown.
Let $S^2=(m+n-2)^{-1}\Big( \sum_i^m (X_i-\bar{X})^2+\sum_j^m (Y_i-\bar{Y})^2 \Big)$
Can anyone help me find the distributions of:
$(m+n-2)S^2/\sigma^2\tag{1}$ And $\frac{\bar{X}-\bar{Y}-(\mu_1-\mu_2)}{\sqrt{S^2(\frac{1}{m}+\frac{1}{n})}}\tag{2}$
For $(2)$ I am having no real luck at all. With $(1)$ I have the obvious step of writing $(m+n-2)S^2/\sigma^2=\frac{1}{\sigma^2}\Big( \sum_i^m (X_i-\bar{X})^2+\sum_j^m (Y_i-\bar{Y})^2 \Big)$ On the RHS is there some connection with the variance of $X$ and $Y$?
$(m+n-2)S^2/\sigma^2=\frac{1}{\sigma^2}\Big( \frac{\alpha^2}{m-1}+\frac{\beta^2}{n-1}\Big)$ Where $\alpha^2,\beta^2$ are the sample variance for $X$ and $Y$ which we can relate to the addition of 2 chi square distributions? i.e. $\chi_{m-1}^2+\chi_{n-1}^2=\chi_{m+n-2}^2$