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The problem states: "Let $X = (X_1, X_2, \dots, X_n)^T$ be a multivariate normal random vector. Show that $X_i$ and $X_j$, $i \neq j$, are independent if and only if $X_i$ and $X_j$ are uncorrelated."

Going from pairwise independence to uncorrelated is easy: Just use the fact that $E(X_i X_j) = E(X_i)E(X_j)$ to show that $\mathrm{Corr}(X_i X_j)=0$. But how to prove the converse? Uncorrelated doesn't imply independence, does it?

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marked as duplicate by Namaste, Community Oct 29 '17 at 19:21

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Hint: Write the joint probability density function for $X_1, \dots, X_n$ given that their correlation matrix is diagonal (they are pairwise uncorrelated). Then try to write the probability density function as a product of the probability density functions for $X_1$ and that of $X_2$, and so on. If you can factor the probability density function, then they are independent.

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