I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?
If two Gaussian random variables are uncorrelated, they are statistically independent
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10You can't prove that because it is not true in general. Uncorrelated _jointly Gaussian_ random variables are independent. If the random variables are Gaussian but not jointly Gaussian, then they could be uncorrelated and yet be dependent. There are standard examples. Search this web site for other answers to this problem. – 2012-01-20
4 Answers
If $X$ and $Y$ are jointly gaussian, and uncorrelated, you can show that $f_{XY}(x,y) = f_X(x)f_Y(y);$ this assures independence.
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0But be sure to check out [this recent question](http://stats.stackexchange.com/q/21429/6633) on stats.SE where a related issue regarding independence is being discussed. – 2012-01-21
The covariance matrix of uncorrelated random variables $X$ and $Y$ is a diagonal matrix. If the two random variables are jointly Gaussian, we can write the joint PDF as
$\frac{1}{\sqrt{(2\pi)^2*|K|}}*e^{-\frac{(x-\mu)^TK^{-1}(x-\mu)}{2}}$ where K is the covariance matrix of X and Y, $K = \begin{bmatrix} \sigma_x^2 & 0 \\ 0 & \sigma_y^2 \end{bmatrix}.$ If we use this $K$ in above equation, we can split the joint pdf into marginal pdf of $X$ and $Y$, hence independence is proved.
Two random variables, X,Y, are said to be uncorrelated if their co-variance, E(XY) − E(X)E(Y), is zero.
This mean : E(XY) = E(X)E(Y)
then , they are statistically independent