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Normally distributed and uncorrelated does not imply independent
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 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 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.
==Examples==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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