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Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$.

Can someone give a hint why this is true ?

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Try using characteristic functions. –  Eckhard Mar 26 '13 at 9:56
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this is actually false... You need $(X,Y)$ to be jointly normal. you only stated marginal distribution –  Lost1 Feb 1 at 2:04

1 Answer 1

what you said is true only when $(X,Y)$ is known to be jointly Gaussian. What you said in the question suggests that you only know $X$ and $Y$ are marginally Gaussian, for which correlation being $0$ is not sufficient!

Consider this construction:

take $X$ to be $N(0,1)$ distribution. Take $Y=X$ if $|X|\leq c$, $Y=-X$, if $|X|>c$.

where $c>0$ is chosen such that

$E [X^2] 1_{\{X\leq c\}}=E [X^2] 1_{\{X> c\}}$

Convince yourself $Y$ is also distributed as $N(0,1)$!

Now note this:

$\rho = E[XY] -E[X]E[Y] =E[X^2]1_{\{X\leq c\}}- E [X^2] 1_{\{X> c\}} -EXEY =0$ but $X$ and $Y$ clearly are not independent

Moral of the story: you always need the joint distribution to be Gaussian! Knowing the marginals is not enough

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