Uncorrelatedness (probability theory)
In probability theory and statistics, two real-valued random variables, formula_1, formula_2, are said to be uncorrelated if their covariance, formula_3, is zero. If two variables are uncorrelated, there is no linear relationship between them. Uncorrelated random variables have a Pearson correlation coefficient of zero, except in the trivial case when either variable has zero variance (is a constant). In this case the correlation is undefined. In general, uncorrelatedness is not the same as orthogonality, except in the special case where at least one of the two random variables has an expected value of 0. In this  case, the covariance is the expectation of the product, and formula_1 and formula_2 are uncorrelated if and only if formula_6. If formula_1 and formula_2 are independent, with finite second moments, then they are uncorrelated. However, not all uncorrelated variables are independent. Definition. Definition for two real random variables. Two random variables formula_9 are called uncorrelated if their covariance formula_10 is zero. Formally: Definition for two complex random variables. Two complex random variables formula_11 are called uncorrelated if their covariance formula_12 and their pseudo-covariance formula_13 is zero, i.e. formula_14 Definition for more than two random variables. A set of two  or more random variables formula_15 is called uncorrelated if each pair of them is uncorrelated. This is equivalent to the requirement that the non-diagonal elements of the autocovariance matrix formula_16 of the random vector formula_17 are all zero. The autocovariance matrix is defined as: The claim is that formula_22 and formula_1 have zero covariance (and thus are uncorrelated), but are not independent. Examples of dependence without correlation. Example 1. Proof: Taking into account that where the second equality holds because formula_1 and formula_2 are independent, one gets Therefore, formula_22 and formula_1 are uncorrelated. Independence of formula_22 and formula_1 means  that for all formula_34 and formula_35, formula_36. This is not true, in particular, for formula_37 and formula_38. Thus formula_41 so formula_22 and formula_1 are not independent. Q.E.D. Example 2. If formula_1 is a continuous random variable uniformly distributed on formula_45 and formula_46, then formula_1 and formula_2 are uncorrelated even though formula_1 determines formula_2 and a particular value of formula_2 can be produced by only one or two values of formula_1 : formula_53 on the other hand, formula_54 is 0 on the triangle defined by formula_55 although formula_56 is not null on this domain. Therefore formula_57 and the variables are not independent . formula_58 where and Uncorrelated stochastic processes. Two stochastic processes formula_61 and formula_62 are called uncorrelated if their cross-covariance formula_63 is zero for all times. Formally: