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

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Bivariate Normal Distribution / Multivariate Normal (Overview)

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Generate Multivariate Random Data in R (2 Examples) | Simulate Variables

Multivariate Experimental Design