22 Feb
2016
22 Feb
'16
4:58 a.m.
in probability theory: variance + mean^2 = meansquare
This is the one-dimensional special case of the Huygens-Steiner theorem, which states: |X - E(X)|^2 + |E(X) - y|^2 = E(|X - y|^2) where X is a random variable on an inner product space and y is an arbitrary point in the inner product space. A generalisation is that: |X - E(X)|^2 + |Y - E(Y)^2| + |E(X) - E(Y)|^2 = E(|X - Y|^2) where X and Y are two independent random variables on the same inner product space. Equivalently: "The mean squared distance between two distributions is equal to the sum of their variances and the squared distance between their barycentres." This can be proved by two applications of Huygens-Steiner. Best wishes, Adam P. Goucher