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Concept
Normally distributed and uncorrelated does not imply independent
Summary
In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables are normally distributed. This article demonstrates that assumption of normal distributions does not have that consequence, although the multivariate normal distribution, including the bivariate normal distribution, does.
To say that the pair of random variables has a bivariate normal distribution means that every linear combination of and for constant (i.e. not random) coefficients and (not both equal to zero) has a univariate normal distribution. In that case, if and are uncorrelated then they are independent. However, it is possible for two random variables and to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.
Suppose has a normal distribution with expected value 0 and variance 1. Let have the Rademacher distribution, so that or , each with probability 1/2, and assume is independent of . Let . Then
and are uncorrelated;
both have the same normal distribution; and
and are not independent.
To see that and are uncorrelated, one may consider the covariance : by definition, it is
Then by definition of the random variables , , and , and the independence of from , one has
To see that has the same normal distribution as , consider
(since and both have the same normal distribution), where is the cumulative distribution function of the Standard normal distribution..
To see that and are not independent, observe that or that .
Finally, the distribution of the simple linear combination concentrates positive probability at 0: . Therefore, the random variable is not normally distributed, and so also and are not jointly normally distributed (by the definition above).
Suppose has a normal distribution with expected value 0 and variance 1. Let
where is a positive number to be specified below.
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