# Joint Density for multivariate normal distribution and and a univariate normal distribution

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?

• This distribution does not have a density, so you have to use some other way to write the distribution. See stats.stackexchange.com/questions/159313 for a short discussion. Related threads include stats.stackexchange.com/questions/63817, stats.stackexchange.com/questions/91045, stats.stackexchange.com/questions/38111, and perhaps other hits from this site search.
– whuber
Commented Nov 27, 2020 at 23:25
• @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? Commented Nov 27, 2020 at 23:54
• Use the basic laws of covariance. (There's nothing special about the distributions being Normal.)
– whuber
Commented Nov 28, 2020 at 15:14