How to determine the distribution of $U:=(X,Y,Z)$?

Question

I've got a question concerning the distribution of a multi dimensional random variable.

I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. How can I then determine the distribution of $U:=(X,Y,Z)$ if I do not certainly know that $X,Y$ and $Z$ are independent?

I know in case that $X,Y$ and $Z$ were independent I could easily find the multivariate distribution by multiplicating the densities $f_X$, $f_Y$ and $f_Z$.

But, as I said, $X,Y$ and $Z$ do not have to be independent. What can I do then?

Answer 1

If $X$, $Y$ and $Z$ are not independent, this means that the covariance matrix $\Sigma$ is not diagonal.

The covariance matrix in your case is defined as follows:

$$\Sigma = \left[\begin{array}{ccc}\mathbb{E}[(X-\mu_X)^2] & \mathbb{E}[(X-\mu_X)(Y-\mu_Y)] & \mathbb{E}[(X-\mu_X)(Z-\mu_Z)]\\ \mathbb{E}[(Y-\mu_Y)(X-\mu_X)] & \mathbb{E}[(Y-\mu_Y)^2] & \mathbb{E}[(Y-\mu_Y)(Z-\mu_Z)]\\ \mathbb{E}[(Z-\mu_Z)(X-\mu_X)] & \mathbb{E}[(Z-\mu_Z)(Y-\mu_Y)] & \mathbb{E}[(Z-\mu_Z)^2]\end{array} \right]$$

where $\mathbb{E}[X] = \mu_X$, $\mathbb{E}[Y] = \mu_Y$ and $\mathbb{E}[Z] = \mu_Z$.

If you have the covariance matrix, then you can use the general formula you find here in order to obtain the joint distribution you are looking for!