Editing Normal distribution (section)

===== Scalar form =====
The following auxiliary formula is useful for simplifying the [[posterior distribution|posterior]] update equations, which otherwise become fairly tedious.

<math display=block>a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac{ay+bz}{a+b}\right)^2 + \frac{ab}{a+b}(y-z)^2</math>

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and [[completing the square]]. Note the following about the complex constant factors attached to some of the terms:
# The factor <math display=inline>\frac{ay+bz}{a+b}</math> has the form of a [[weighted average]] of ''y'' and ''z''.
# <math display=inline>\frac{ab}{a+b} = \frac{1}{\frac{1}{a}+\frac{1}{b}} = (a^{-1} + b^{-1})^{-1}.</math> This shows that this factor can be thought of as resulting from a situation where the [[Multiplicative inverse|reciprocals]] of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the [[harmonic mean]], so it is not surprising that <math display=inline>\frac{ab}{a+b}</math> is one-half the [[harmonic mean]] of ''a'' and ''b''.