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)
“The brotherhood of men does not imply their equality. Families have their fools and their men of genius, their black sheep and their saints, their worldly successes and their worldly failures. A man should treat his brothers lovingly and with justice, according to the deserts of each. But the deserts of every brother are not the same.”
—Aldous Huxley (1894–1963)
“The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.”
—Charles Sanders Peirce (1839–1914)