...Or, more rigorously, non-correlation does not imply independence.

As this little guy and everybody else knows, one of the most famous correlation coefficients out there is Pearson's correlation coefficient: cor(ξ, η) = (E[(ξ-E[ξ])(η-E[η])])/sqrt(D[ξ]D[η]), where E[x] is the mathematical expectation of random variable x, D[x] is the dispersion of random variable x, and sqrt(x) is the (prime) square root of x.

As we all know, if cor(ξ, η) != 0, then ξ and η are not independent random variables. But recently, this little guy heard that it does not follow from cor(ξ, η) = 0 that ξ and η are independent. Obviously, he craves the light of knowledge and wants to hear some examples of non-independent random variables having a correlation coefficient of 0.

  • Tomorrow_Farewell [any, they/them]
    ·
    1 month ago

    I'll start.

    Let's consider two perfectly fair three-sided coins, one with the faces labelled '-1', '0', '1', and the other - with the faces labelled '0', '1', '0'. When we conduct the following experiment, ξ will be the random variable that maps the label of the face of the first coin that we got to a real number, and η does the same for the second one.

    The experiment is as follows: we toss the first coin, and, after it lands, we set the second coin to some face chosen as follows:

    • If the first coin lands on '-1' or '1', we set the second one to one of its '0' faces.
    • If the first coin lands on '0', we set the second coin to '1'.

    As such, E[ξ] = -1*1/3+0*1/3+1*1/3 = 0, E[η] = 0*1/3+1*1/3+0*1/3 = 1/3, covariance cov(ξ, η) = E[(ξ-E[ξ])(η-E[η])] = (-1-0)(0-1/3)*1/3+(0-0)(1-1/3)*1/3+(1-0)(0-1/3)*1/3 = ((-1-0)-(1-0))(0-1/3)*1/3 = 0, and, thus, cor(ξ, η) = 0/sqrt(D[ξ]D[η]) = 0.
    However, it's obvious that the random variables are not independent, as P(ξ = 0, η = 1) = 1/3 != 1/9 = P(ξ = 0)*P(η = 1).