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A~N(0,1) and I~Bernoulli(1/2) are independent random variables. Then both X1=A and X2=(1-2I)*A are Gaussian random variables. They even have 0 covariance but th
by leni536 7y ago
A~N(0,1) and I~Bernoulli(1/2) are independent random variables. Then both X1=A and X2=(1-2I)*A are Gaussian random variables. They even have 0 covariance but they are not independent. Also X1+X2 is not Gaussian.
- kgwgk 7y agoYou were right on the “or there should be some other criteria” part, but not on the “They need to be independent” part. If you start with independent Gaussians A and B and accept that A+B and A-B are Gaussians you’ll easily see that the sum of those is also Gaussian (because it’s 2A) even though they are not independent.