Editing Harmonic mean (section)

==Beta distribution==
{{Unreferenced section|date=December 2019}}
[[File:Harmonic mean for Beta distribution for alpha and beta ranging from 0 to 5 - J. Rodal.jpg|thumb|Harmonic mean for Beta distribution for 0 < α < 5 and 0 < β < 5]]
[[File:(Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2 - J. Rodal.jpg|thumb|(Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2]]
[[File: Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front - J. Rodal.jpg|thumb|Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front]]
[[File: Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front - J. Rodal.jpg|thumb|Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front]]

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

:<math>H = \frac{\alpha - 1}{\alpha + \beta - 1} \text{ conditional on } \alpha > 1 \, \, \& \, \, \beta > 0 </math>

The harmonic mean with ''α'' < 1 is undefined because its defining expression is not bounded in [0, 1].

Letting ''α'' = ''β''

: <math>H = \frac{\alpha - 1}{2 \alpha - 1}</math>

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:

:<math>\begin{align}
  \lim_{\alpha \to 0} H &= \text{ undefined } \\
  \lim_{\alpha \to 1} H &= \lim_{\beta \to \infty}  H = 0 \\
  \lim_{\beta \to 0}  H &= \lim_{\alpha \to \infty} H = 1
\end{align}</math>

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

A second harmonic mean (''H''<sub>1 − X</sub>) also exists for this distribution

:<math>H_{1-X} =  \frac{\beta - 1}{\alpha + \beta - 1} \text{ conditional on } \beta > 1 \, \, \& \, \, \alpha > 0</math>

This harmonic mean with ''β'' < 1 is undefined because its defining expression is not bounded in [ 0, 1 ].

Letting ''α'' = ''β'' in the above expression

:<math>H_{1-X} = \frac{\beta - 1}{2 \beta - 1} </math>

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:

: <math>\begin{align}
  \lim_{\beta \to 0}  H_{1-X} &= \text{ undefined } \\
  \lim_{\beta \to 1}  H_{1-X} &= \lim_{\alpha \to \infty} H_{1-X} = 0 \\
  \lim_{\alpha \to 0} H_{1-X} &= \lim_{\beta \to \infty}  H_{1-X} = 1
\end{align}</math>

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