I have the following relation between random variables: $$x^i_{t+1} = \sum_{j=0,j \ne i}^{N}a_{ij}x^j_t$$ where $x_t^j$ follows $N(\mu_j,\sigma_j)$. Assume $x_t^j$ are the independent variables and $x^i_{t+1}$ is the dependent variable. I am trying to find the joint probability distribution of independent variables and the depedent one. That is $P(x^i_{t+1}, x_t)$, where $x_t$ follows $N(\mu,\sigma)$, $\mu=[\mu_1,\dots,\mu_{N-1}]^T$, $\sigma= diag[\sigma_1,\dots,\sigma_{N-1}]$ since all independent variables are statistically independent on each other. I know the joint distribution should be a Gaussian but I am not sure how to write a joint distribution when the bivariate Gaussian parameters have different dimensionality?



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