Uncorrelated Data with Bivariate Normal Distribution is Independent

Theorem

Let $X$ and $Y$ be random variables with bivariate normal distribution.

Let $X$ and $Y$ be uncorrelated.

Then $X$ and $Y$ be independent.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): correlation coefficient: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): correlation coefficient: 1.