Harmonic Mean - Beta Distribution

Beta Distribution

The harmonic mean of a beta distribution with shape parameters α and β is:

The harmonic mean with α < 1 is undefined because its defining expression is not bounded in .

Letting α = β

showing that for α = β the harmonic mean ranges from 0 for α = β = 1, to 1/2 for α = β → ∞.

The following are the limits with one parameter finite (non zero) and the other parameter approaching these limits:

With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.

A second harmonic mean (H_{1 - X}) also exists for this distribution

This harmonic mean with β < 1 is undefined because its defining expression is not bounded in .

Letting α = β in the above expression

showing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.

The following are the limits with one parameter finite (non zero) and the other approaching these limits:

Although both harmonic means are asymmetric, when α = β the two means are equal.