I am struggling with writing down a relationship between two variables and I am happy for each and every comment that points out, how I can overcome this problem:
Let $x\sim N(\mu,\Sigma), x \in\mathbb{R}^{N-1}$ be a vector that is multivariate normally distributed with mean parameter $\mu$ and covariance matrix $\Sigma$. $x_N$ is of such that $\sum_{i=1}^{N}x_i=1$. (Therefore: conditional on $x$ the scalar $x_N$ is deterministic defined as $1-\sum x$.) It is a standard result that $X_n$ (unconditional of $x$) as the sum of normal distributed variables is also normally distributed with mean parameter $1-\iota'x$ (where $\iota$ is a vector of ones) and variance $\iota'\Sigma\iota$. However, it is an obstacle for me to write down the joint distribution of $z:=(x,x_N)$ (which is apparently not normal due to the deterministic behavior of $x_N$ given $x$). However, I can easily sample from the joint distribution and it is not surprising to see for me that the expected value of $z=E(z)=(\mu,1-\iota'\mu)$. Sampling also gives me that the covariance between $x_N$ and $x_i$ ($i\in\{1,\ldots,N-1\}$) is what I would expect: $$cov(x_N,x_i)=cov(1-\sum_{j=1}^{N-1}x_j,x_i)=-\sum_{j=1}^{N-1}cov(x_j,x_i)$$ which is nothing but minus the sum of the $i$-th row of $\Sigma$. Can anyone help me to explain how to mathematically show that this is correct although I don't have the joint distribution of $z$?