In probability theory, two random variables being uncorrelated does not imply their independence. In some contexts, uncorrelatedness implies at least pairwise independence (as when the random variables involved have Bernoulli distributions).
It is sometimes mistakenly thought that one context in which uncorrelatedness implies independence is when the random variables involved are normally distributed. However, this is incorrect if the variables are merely marginally normally distributed but not jointly normally distributed.
Suppose two random variables X and Y are jointly normally distributed. That is the same as saying that the random vector (X, Y) has a multivariate normal distribution. It means that the joint probability distribution of X and Y is such that 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 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.
Famous quotes containing the words distributed, imply and/or independent:
“Taking food alone tends to make one hard and coarse. Those accustomed to it must lead a Spartan life if they are not to go downhill. Hermits have observed, if for only this reason, a frugal diet. For it is only in company that eating is done justice; food must be divided and distributed if it is to be well received.”
—Walter Benjamin (1892–1940)
“I do not mean to imply that the good old days were perfect. But the institutions and structure—the web—of society needed reform, not demolition. To have cut the institutional and community strands without replacing them with new ones proved to be a form of abuse to one generation and to the next. For so many Americans, the tragedy was not in dreaming that life could be better; the tragedy was that the dreaming ended.”
—Richard Louv (20th century)
“Milton’s the prince of poets—so we say;
A little heavy, but no less divine:
An independent being in his day—”
—George Gordon Noel Byron (1788–1824)