# Deriving Wilk's distribution for two normal independent variables

Let $X$ and $Y$ be two independent, normal random variables with known and equal variances, let $\{X_1,\ldots,X_m\}$ and $\{Y_1,\ldots,Y_n\}$ be random samples of size m and n respectively. We are testing $H_0:\mu_X=\mu_Y=\mu$, $H_1:\mu_X\neq\mu_Y$.
The Likelihood Ratio ($\frac{H_0}{H_A}$) is
$$\Lambda=\dfrac{\Bigg(\dfrac{1}{2\pi\sigma^2}\Bigg)^{(m+n)/2}\text{exp}\Bigg(-\dfrac{1}{2\sigma^2}\Bigg[\displaystyle\sum_{i=1}^m(x_i-\hat{\mu})^2+\sum_{j=1}^n(y_j-\hat{\mu})^2\Bigg]\Bigg)}{\Bigg(\dfrac{1}{2\pi\sigma^2}\Bigg)^{(m+n)/2}\text{exp}\Bigg(-\dfrac{1}{2\sigma^2}\Bigg[\displaystyle\sum_{i=1}^m(x_i-\hat{\mu}_X)^2+\sum_{j=1}^n(y_j-\hat{\mu}_Y)^2\Bigg]\Bigg)}$$

The ML estimators are
$$\hat{\mu}_X=\cfrac{1}{m}\displaystyle\sum_{i=1}^mx_i$$ $$\hat{\mu}_Y=\cfrac{1}{n}\displaystyle\sum_{j=1}^ny_j$$ $$\hat{\mu}=\cfrac{1}{m+n}\displaystyle\Bigg(\sum_{i=1}^mx_i+\sum_{j=1}^ny_j\Bigg)$$ We need to: derive the Likelihood Ratio $\Lambda$ and the distribution of $−2log\Lambda$.
The distribution should be similar to the two sample z test for comparing two means.