Editing Beta distribution (section)

====Harmonic mean====
[[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&nbsp;<&nbsp;''α''&nbsp;<&nbsp;5 and 0&nbsp;<&nbsp;''β''&nbsp;<&nbsp;5]]
[[File:(Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2 - J. Rodal.jpg|thumb|Harmonic mean for beta distribution versus ''α'' and ''β'' 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&nbsp;−&nbsp;''X''), smaller values ''α'' and ''β'' 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&nbsp;−&nbsp;''X''), larger values ''α'' and ''β'' in front]]

The inverse of the [[harmonic mean]] (''H<sub>X</sub>'') of a distribution with [[random variable]] ''X'' is the arithmetic mean of 1/''X'', or, equivalently, its expected value. Therefore, the [[harmonic mean]] (''H<sub>X</sub>'') of a beta distribution with shape parameters ''α'' and ''β'' is:

:<math> \begin{align}
H_X &= \frac{1}{\operatorname{E}\left[\frac{1}{X}\right]} \\
    &=\frac{1}{\int_0^1 \frac{f(x;\alpha,\beta)}{x}\,dx} \\
    &=\frac{1}{\int_0^1 \frac{x^{\alpha-1}(1-x)^{\beta-1}}{x \Beta(\alpha,\beta)}\,dx} \\
    &= \frac{\alpha - 1}{\alpha + \beta - 1}\text{ if } \alpha > 1 \text{ and }  \beta > 0 \\
\end{align}</math>

The [[harmonic mean]] (''H<sub>X</sub>'') of a beta distribution with ''α'' < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter ''α'' less than unity.

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

:<math>H_X = \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'' → ∞.

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

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

The harmonic mean plays a role in maximum likelihood estimation for the four parameter case, in addition to the geometric mean. Actually, when performing maximum likelihood estimation for the four parameter case, besides the harmonic mean ''H<sub>X</sub>'' based on the random variable ''X'', also another harmonic mean appears naturally: the harmonic mean based on the linear transformation (1&nbsp;−&nbsp;''X''), the mirror-image of ''X'', denoted by ''H''<sub>1&nbsp;−&nbsp;''X''</sub>:

:<math>H_{1-X} = \frac{1}{\operatorname{E} \left[\frac 1 {1-X}\right]} = \frac{\beta - 1}{\alpha + \beta-1} \text{ if } \beta > 1, \text{ and } \alpha> 0. </math>

The [[harmonic mean]] (''H''<sub>(1&nbsp;−&nbsp;''X'')</sub>) of a beta distribution with ''β'' < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter ''β'' less than unity.

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

:<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'' → ∞.

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{ is 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 ''H''<sub>''X''</sub> and  ''H''<sub>1−''X''</sub> are asymmetric, in the case that both shape parameters are equal ''α'' = ''β'', the harmonic means are equal: ''H''<sub>''X''</sub> = ''H''<sub>1−''X''</sub>.  This equality follows from the following symmetry displayed between both harmonic means:

:<math>H_X (\Beta(\alpha, \beta) )=H_{1-X}(\Beta(\beta, \alpha) ) \text{ if } \alpha, \beta> 1.</math>