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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This distribution does not have a density, so you have to use some other way to write the distribution. See https://stats.stackexchange.com/questions/159313 for a short discussion. Related threads include https://stats.stackexchange.com/questions/63817, https://stats.stackexchange.com/questions/91045, https://stats.stackexchange.com/questions/38111, and perhaps other hits from [this site search](https://stats.stackexchange.com/search?q=singular+normal+distribution). – whuber Nov 27 '20 at 23:25
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@whuber I understand from the links that the joint distribution is represented in a lower dimensional space, which is concluded from the fact that the covariance matrix is singular. However, I don't understand how to obtain the covariance matrix of the joint? – rando Nov 27 '20 at 23:54
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1Use the basic laws of covariance. (There's nothing special about the distributions being Normal.) – whuber Nov 28 '20 at 15:14