Independence, uncorrelatedness, and coprimeness (2018-06-29)

What do these three have in common? Remember that two random variables \(X\) and \(Y\) is called independent if \(P(XY) = P(X)P(Y).\) More precisely, let \(F_X(x)\) be \(X\)'s CDF and \(F_Y(y)\) \(Y\)'s CDF. \(X\) and \(Y\) are independent iff\begin{align}F_{X, Y}(x, y) = F_X(x) F_Y(y),\end{align}for all \(x, y\) where \(F_{X, Y}(x, y)\) is the joint CDF of the combined random variable \((x, y)\).

Next, how about the definition of uncorrelatedness? Two random variables \(X\) and \(Y\) are said to be uncorrelated when their covariance is zero i.e. \(\mathrm{cov}(X, Y) = 0.\) In other words,\begin{align}\mathop{\mathbb{E}}[XY] = \mathop{\mathbb{E}}[X] \mathop{\mathbb{E}}[Y].\end{align}

Again, how to define the coprimeness property of two (positive) intergers? Although it is usually defined by the statement that their greatest common divisor is 1, there is an alternative definition using the divisor function: \(m\) and \(n\) are coprime iff\begin{align}d(m n) = d(m) d(n),\end{align}where \(d(m)\) is the number of divisors of \(m\).

Now we have seen that the above three notions can be defined by some equation of form \(f(xy) = f(x)f(y)\) with a certain function \(f\).

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