# Distribution of $(Y_1,Y_2)^\mathsf{T}$ where $Y_i=(\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}X_i$

Let $$X_1$$ and $$X_2$$ be two vectors mutually independent such that $$X_i\sim\mathcal{N}_p(\mu_i,\Sigma)$$ with $$\mu_i$$ a vector $$\in\mathbb R^p$$ and the covariance matrix $$\Sigma\in\mathbb R^{p\times p}$$. Let $$Y_i=(\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}X_i$$. Which is the joint distribution of the vector $$(Y_1,Y_2)^\mathsf{T}$$? Are the components independent?

First I get that each $$Y_i$$ is univariate normal where $$Y_1\sim\mathcal N\Big((\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}\mu_1,(\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}(\mu_1-\mu_2)\Big)$$ $$Y_2\sim\mathcal N\Big((\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}\mu_2,(\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}(\mu_1-\mu_2)\Big)$$

Then \begin{align} \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix} \sim \mathcal{N}_2 \left( \begin{bmatrix} (\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}\mu_1 \\ (\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}\mu_2 \end{bmatrix}, \begin{bmatrix} (\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}(\mu_1-\mu_2)) & 0\\ 0 & (\mu_1-\mu_2)^\mathsf{T}\Sigma^{-1}(\mu_1-\mu_2)) \end{bmatrix} \right). \end{align}

Since $$X_1$$ and $$X_2$$ are mutually independent, then $$Y_1$$ and $$Y_2$$ are also independent because they are linear combinations of each of the vectors, so $$\mathrm{Cov}\,(Y_1,Y_2)=0$$. However the components of $$(Y_1,Y_2)^\mathsf{T}$$ are not independent because they have the same covariance matrix and $$\mathrm{Cov}=0$$ only implies in independence under multivariate normality.

Is it right?

You are correct that $Y_1$ and $Y_2$ are independent since $Y_i$ is a function of $X_i$ and the $X_i$ are independent. Your calculations of the distributions look right too.
However when you say the components of $(Y_1,Y_2)^T$ are not independent this does not make sense. $Y_1$ and $Y_2$ are the components of this vector, and as you said, they are independent. Not sure if this is what's causing the confusion here, but note that $Y_1$ and $Y_2$ are scalars. $(\mu_1-\mu_2)^T\Sigma^{-1}(\mu_1-\mu_2)$ is a number... the variance of $Y_1$ (and also of $Y_2$).