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Administrator
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Hi X,
Gujarati is referring to the following:
Suppose two random variables X and Y are jointly normally distributed. In other words, the random vector (X, Y) has a multivariate normal distribution. Here is a way to describe multivariate normal: We say that the random vector (X, Y) has a multivariate normal distribution if joint probability distribution of X and Y is such that each linear combination of X and Y is normally distributed, i.e. for any two constant (i.e., non-random) scalars a and b, the random variable aX + bY is normally distributed. In that case if X and Y are uncorrelated, i.e., their covariance cov(X, Y) is zero, then they are independent.
However, it is possible for two random variables X and Y to have marginal normal distributions each but their joint distribution is not multivariate normal. In such cases, there is a possibility that they are uncorrelated, but not independent. For example:
Suppose X has a normal distribution with expected value 0 and variance 1. Let W have the Rademacher distribution, so that W = 1 or −1, each with probability 1/2, and assume W is independent of X. Let Y = WX. Clearly, X and Y have normal distribution and are uncorrelated. (For working refer the wiki link you provided)
But (X, Y) does not have a multivariate normal distribution because
X + Y = 2X if W = 1
and X + Y = 0 if W = -1 is not normally distributed (since X + Y takes value 0 with probability 1/2)
To summarize,
(X, Y) has a multivariate normal distribution plus zero covariance between X and Y implies independence of X and Y
However, (X, Y) has some joint distribution such that marginal distributions of both X and Y are normal, then zero covariance between X and Y does not imply independence of X and Y.
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