If variables with a multivariate normal distribution have covariances that equal zero, then: the correlation will most often be zero, but does not have to be. the marginal distribution of each of the variables is no longer normal: you should use the X2 distribution to calculate probabilities. the variables are independent:

The Normal Distribution

Multivariate Gamma Distributions

Bivariate Normal Distribution / Multivariate Normal (Overview)

The Multivariate Gaussian distribution isn't as scary as you think

Consider sampling from a multivariate normal distribution with ... | Filo

Python SciPy Stats Multivariate_Normal

Multivariate Experimental Design