I am given that
joint normal distribution factorizes if the covariance matrix is diagonal
I was wondering why such a fact holds true. Is there a rigorous way to show that? Any reference would be extremely appreciated.
2026-04-03 16:02:55.1775232175
Joint normal distribution factorizes if covariance matrix is diagonal. Why?
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Jointly normal distributions are parametrized by their means and covariance matrices (think about characteristic function of a normal random variable: the only two parameters involved are exactly mean and variance). Independent random variables have covariance $0$: conversely, jointly normal random variables with covariance $0$ have the same mean and covariance as independent normal random variables, and therefore the same distribution.