Editing Beta distribution (section)

===Measures of central tendency===

====Mode====
The [[Mode (statistics)|mode]] of a beta distributed [[random variable]] ''X'' with ''α'', ''β'' > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:<ref name=JKB>{{cite book|last1=Johnson|first1= Norman L. |first2= Samuel|last2= Kotz |first3= N. |last3= Balakrishnan| year=1995 |title=Continuous Univariate Distributions Vol. 2 |edition=2nd |publisher= Wiley |isbn= 978-0-471-58494-0 |chapter= Chapter 25: Beta Distributions}}</ref>

:<math>\frac{\alpha - 1} {\alpha + \beta - 2} .</math>

When both parameters are less than one (''α'', ''β'' < 1), this is the anti-mode: the lowest point of the probability density curve.<ref name=Wadsworth>{{cite book|last=Wadsworth |first=George P. and Joseph Bryan |title=Introduction to Probability and Random Variables|url=https://archive.org/details/introductiontopr0000wads |url-access=registration |year=1960|publisher=McGraw-Hill}}</ref>

Letting ''α'' = ''β'', the expression for the mode simplifies to 1/2, showing that for ''α'' = ''β'' > 1 the mode (resp. anti-mode when {{nowrap|''α'', ''β'' < 1}}), is at the center of the distribution: it is symmetric in those cases.  See [[Beta distribution#Shapes|Shapes]] section in this article for a full list of mode cases, for arbitrary values of ''α'' and ''β''. For several of these cases, the maximum value of the density function occurs at one or both ends.  In some cases the (maximum) value of the density function occurring at the end is finite. For example, in the case of ''α'' = 2, ''β'' = 1 (or ''α'' = 1, ''β'' = 2), the density function becomes a [[Triangular distribution|right-triangle distribution]] which is finite at both ends. In several other cases there is a [[Mathematical singularity|singularity]] at one end, where the value of the density function approaches infinity. For example, in the case ''α'' = ''β'' = 1/2, the beta distribution simplifies to become the [[arcsine distribution]].  There is debate among mathematicians about some of these cases and whether the ends (''x'' = 0, and ''x'' = 1) can be called ''modes'' or not.<ref name="Handbook of Beta Distribution" /><ref name="Mathematical Statistics with MATHEMATICA">{{cite book |last1=Rose |first1=Colin |last2=Smith |first2=Murray D. |title=Mathematical Statistics with MATHEMATICA |year=2002 |publisher=Springer |isbn=978-0387952345}}</ref>
[[File:Mode Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px|thumb|Mode for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ β ≤ 5]]

* Whether the ends are part of the [[Domain of a function|domain]] of the density function
* Whether a [[Mathematical singularity|singularity]] can ever be called a ''mode''
* Whether cases with two maxima should be called ''bimodal''

====Median====
[[File:Median Beta Distribution for alpha and beta from 0 to 5 - J. Rodal.jpg|325px|thumb|Median for beta distribution for 0 ≤ ''α'' ≤ 5 and 0 ≤ ''β'' ≤ 5]]
[[File:(Mean - Median) for Beta distribution versus alpha and beta from 0 to 2 - J. Rodal.jpg|thumb|(Mean–median) for beta distribution versus alpha and beta from 0 to 2]]

The median of the beta distribution is the unique real number <math>x = I_{1/2}^{[-1]}(\alpha,\beta)</math> for which the [[regularized incomplete beta function]] <math>I_x(\alpha,\beta) = \tfrac{1}{2} </math>. There is no general [[closed-form expression]] for the [[median]] of the beta distribution for arbitrary values of ''α'' and ''β''.  [[Closed-form expression]]s for particular values of the parameters ''α'' and ''β'' follow:{{citation needed|date=February 2013}}

* For symmetric cases  ''α'' = ''β'', median = 1/2.
* For ''α'' = 1 and ''β'' > 0, median <math> =1-2^{-1/\beta}</math> (this case is the [[mirror image|mirror-image]] of the power function [0,1] distribution)
* For ''α'' > 0 and ''β'' = 1, median = <math>2^{-1/\alpha}</math> (this case is the power function [0,1] distribution<ref name="Handbook of Beta Distribution" />)
* For ''α'' = 3 and ''β'' = 2, median = 0.6142724318676105..., the real solution to the [[Quartic function|quartic equation]] 1 − 8''x''<sup>3</sup> + 6''x''<sup>4</sup> = 0, which lies in [0,1].
* For ''α'' = 2 and ''β'' = 3, median = 0.38572756813238945... = 1−median(Beta(3, 2))

The following are the limits with one parameter finite (non-zero) and the other approaching these limits:{{citation needed|date=February 2013}}

:<math> \begin{align}
\lim_{\beta \to  0} \text{median}= \lim_{\alpha \to \infty} \text{median} = 1,\\
\lim_{\alpha\to  0} \text{median}= \lim_{\beta \to  \infty} \text{median} = 0.
\end{align}</math>

A reasonable approximation of the value of the median of the beta distribution, for both α and β greater or equal to one, is given by the formula<ref name=Kerman2011/>

:<math>\text{median} \approx \frac{\alpha - \tfrac{1}{3}}{\alpha + \beta - \tfrac{2}{3}} \text{ for } \alpha, \beta \ge 1.</math>

When ''α'', ''β'' ≥ 1, the [[relative error]] (the [[approximation error|absolute error]] divided by the median) in this approximation is less than 4% and for both ''α'' ≥ 2 and ''β'' ≥ 2 it is less than 1%. The [[approximation error|absolute error]] divided by the difference between the mean and the mode is similarly small:

[[File:Relative Error for Approximation to Median of Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px|Abs[(Median-Appr.)/Median] for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ ''β'' ≤ 5]][[File:Error in Median Apprx. relative to Mean-Mode distance for Beta Distribution with alpha and beta from 1 to 5 - J. Rodal.jpg|325px|Abs[(Median-Appr.)/(Mean-Mode)] for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ ''β'' ≤ 5]]

====Mean====
[[File:Mean Beta Distribution for alpha and beta from 0 to 5 - J. Rodal.jpg|325px|thumb|Mean for beta distribution for {{nowrap|0 ≤ ''α'' ≤ 5}} and {{nowrap|0 ≤ ''β'' ≤ 5}}]]
The [[expected value]] (mean) (''μ'') of a beta distribution [[random variable]] ''X'' with two parameters ''α'' and ''β'' is a function of only the ratio ''β''/''α'' of these parameters:<ref name=JKB />

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

Letting {{nowrap|1=''α'' = ''β''}} in the above expression one obtains {{nowrap|1=''μ'' = 1/2}}, showing that for {{nowrap|1=''α'' = ''β''}} the mean is at the center of the distribution: it is symmetric. Also, the following limits can be obtained from the above expression:

:<math> \begin{align}
\lim_{\frac{\beta}{\alpha} \to  0} \mu = 1\\
\lim_{\frac{\beta}{\alpha} \to  \infty} \mu = 0
\end{align}</math>

Therefore, for ''β''/''α'' → 0, or for ''α''/''β'' → ∞, the mean is located at the right end, {{nowrap|1=''x'' = 1}}. For these limit ratios, the beta distribution becomes a one-point [[degenerate distribution]] with a [[Dirac delta function]] spike at the right end, {{nowrap|1=''x'' = 1}}, with probability&nbsp;1, and zero probability everywhere else. There is 100% probability (absolute certainty) concentrated at the right end, {{nowrap|1=''x'' = 1}}.

Similarly, for ''β''/''α'' → ∞, or for ''α''/''β'' → 0, the mean is located at the left end, {{nowrap|1=''x'' = 0}}.  The beta distribution becomes a 1-point [[Degenerate distribution]] with a [[Dirac delta function]] spike at the left end, ''x'' = 0, with probability 1, and zero probability everywhere else. There is 100% probability (absolute certainty) concentrated at the left end, ''x'' = 0. Following are the limits with one parameter finite (non-zero) and the other approaching these limits:

:<math> \begin{align}
\lim_{\beta \to  0} \mu = \lim_{\alpha \to  \infty} \mu = 1\\
\lim_{\alpha\to  0} \mu = \lim_{\beta \to  \infty} \mu = 0
\end{align}</math>

While for typical unimodal distributions (with centrally located modes, inflexion points at both sides of the mode, and longer tails) (with Beta(''α'',&nbsp;''β'') such that {{nowrap|''α'', ''β'' > 2}}) it is known that the sample mean (as an estimate of location) is not as [[Robust statistics|robust]] as the sample median, the opposite is the case for uniform or "U-shaped" bimodal distributions (with Beta(''α'',&nbsp;''β'') such that {{nowrap|''α'', ''β'' ≤ 1}}), with the modes located at the ends of the distribution.  As Mosteller and Tukey remark (<ref name=MostellerTukey>{{cite book|last=Mosteller|first=Frederick and John Tukey|title=Data Analysis and Regression: A Second Course in Statistics|url=https://archive.org/details/dataanalysisregr0000most|url-access=registration|year=1977|publisher=Addison-Wesley Pub. Co.|isbn=978-0201048544|bibcode=1977dars.book.....M}}</ref> p.&nbsp;207) "the average of the two extreme observations uses all the sample information. This illustrates how, for short-tailed distributions, the extreme observations should get more weight."  By contrast, it follows that the median of "U-shaped" bimodal distributions with modes at the edge of the distribution (with Beta(''α'',&nbsp;''β'') such that {{nowrap|''α'', ''β'' ≤ 1}}) is not robust, as the sample median drops the extreme sample observations from consideration.  A practical application of this occurs for example for [[random walk]]s, since the probability for the time of the last visit to the origin in a random walk is distributed as the [[arcsine distribution]] Beta(1/2,&nbsp;1/2):<ref name=Feller/><ref name=WillyFeller1>{{cite book |last=Feller |first=William |title=An Introduction to Probability Theory and Its Applications |volume=1 |edition=3rd |year=1968 |publisher=Wiley |isbn=978-0471257080}}</ref> the mean of a number of [[realization (probability)|realizations]] of a random walk is a much more robust estimator than the median (which is an inappropriate sample measure estimate in this case).

====Geometric mean====
[[File:(Mean - GeometricMean) for Beta Distribution versus alpha and beta from 0 to 2 - J. Rodal.jpg|thumb|(Mean − GeometricMean) for beta distribution versus ''α'' and ''β'' from 0 to 2, showing the asymmetry between ''α'' and ''β'' for the geometric mean]]
[[File:Geometric Means for Beta distribution Purple=G(X), Yellow=G(1-X), smaller values alpha and beta in front - J. Rodal.jpg|thumb|Geometric means for beta distribution Purple = ''G''(''x''), Yellow = ''G''(1&nbsp;−&nbsp;''x''), smaller values ''α'' and ''β'' in front]]
[[File:Geometric Means for Beta distribution Purple=G(X), Yellow=G(1-X), larger values alpha and beta in front - J. Rodal.jpg|thumb|Geometric means for beta distribution. purple = ''G''(''x''), yellow = ''G''(1&nbsp;−&nbsp;''x''), larger values ''α'' and ''β'' in front]]

The logarithm of the [[geometric mean]] ''G<sub>X</sub>'' of a distribution with [[random variable]] ''X'' is the arithmetic mean of ln(''X''), or, equivalently, its expected value:

:<math>\ln G_X = \operatorname{E}[\ln X]</math>

For a beta distribution, the expected value integral gives:

:<math>\begin{align}
\operatorname{E}[\ln X]
&= \int_0^1 \ln x\, f(x;\alpha,\beta)\,dx \\[4pt]
&= \int_0^1 \ln x \,\frac{ x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}\,dx \\[4pt]
&= \frac{1}{\Beta(\alpha,\beta)} \, \int_0^1 \frac{\partial x^{\alpha-1}(1-x)^{\beta-1}}{\partial \alpha}\,dx \\[4pt]
&= \frac{1}{\Beta(\alpha,\beta)} \frac{\partial}{\partial \alpha} \int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx \\[4pt]
&= \frac{1}{\Beta(\alpha,\beta)} \frac{\partial \Beta(\alpha,\beta)}{\partial \alpha} \\[4pt]
&= \frac{\partial \ln \Beta(\alpha,\beta)}{\partial \alpha} \\[4pt]
&= \frac{\partial \ln \Gamma(\alpha)}{\partial \alpha} - \frac{\partial \ln \Gamma(\alpha + \beta)}{\partial \alpha} \\[4pt]
&= \psi(\alpha) - \psi(\alpha + \beta)
\end{align}</math>

where ''ψ'' is the [[digamma function]].

Therefore, the geometric mean of a beta distribution with shape parameters ''α'' and ''β'' is the exponential of the digamma functions of ''α'' and ''β'' as follows:

:<math>G_X =e^{\operatorname{E}[\ln X]}= e^{\psi(\alpha) - \psi(\alpha + \beta)}</math>

While for a beta distribution with equal shape parameters ''α'' = ''β'', it follows that skewness = 0 and mode = mean = median = 1/2, the geometric mean is less than 1/2: {{nowrap|0 < ''G''<sub>''X''</sub> < 1/2}}.  The reason for this is that the logarithmic transformation strongly weights the values of ''X'' close to zero, as ln(''X'') strongly tends towards negative infinity as ''X'' approaches zero, while ln(''X'') flattens towards zero as {{nowrap|''X'' → 1}}.

Along a line {{nowrap|1=''α'' = ''β''}}, the following limits apply:

:<math> \begin{align}
&\lim_{\alpha = \beta \to 0} G_X = 0 \\
&\lim_{\alpha = \beta \to \infty} G_X =\tfrac{1}{2}
\end{align}</math>

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

:<math> \begin{align}
\lim_{\beta \to  0} G_X = \lim_{\alpha \to  \infty} G_X = 1\\
\lim_{\alpha\to  0} G_X = \lim_{\beta \to  \infty} G_X = 0
\end{align}</math>

The accompanying plot shows the difference between the mean and the geometric mean for shape parameters ''α'' and ''β'' from zero to 2.  Besides the fact that the difference between them approaches zero as ''α'' and ''β'' approach infinity and that the difference becomes large for values of ''α'' and ''β'' approaching zero, one can observe an evident asymmetry of the geometric mean with respect to the shape parameters ''α'' and ''β''. The difference between the geometric mean and the mean is larger for small values of ''α'' in relation to ''β'' than when exchanging the magnitudes of ''β'' and ''α''.

[[Norman Lloyd Johnson|N. L.Johnson]] and [[Samuel Kotz|S. Kotz]]<ref name=JKB /> suggest the logarithmic approximation to the digamma function ''ψ''(''α'') ≈ ln(''α''&nbsp;−&nbsp;1/2) which results in the following approximation to the geometric mean:

:<math>G_X \approx \frac{\alpha \, - \frac{1}{2}}{\alpha +\beta - \frac{1}{2}}\text{ if } \alpha, \beta > 1.</math>

Numerical values for the [[relative error]] in this approximation follow: [{{nowrap|1=(''α'' = ''β'' = 1): 9.39%}}]; [{{nowrap|1=(''α'' = ''β'' = 2): 1.29%}}];  [{{nowrap|1=(''α'' = 2, ''β'' = 3): 1.51%}}];  [{{nowrap|1=(''α'' = 3, ''β'' = 2): 0.44%}}]; [{{nowrap|1=(''α'' = ''β'' = 3): 0.51%}}]; [{{nowrap|1=(''α'' = ''β'' = 4): 0.26%}}]; [{{nowrap|1=(''α'' = 3, ''β'' = 4): 0.55%}}];  [{{nowrap|1=(''α'' = 4, ''β'' = 3): 0.24%}}].

Similarly, one can calculate the value of shape parameters required for the geometric mean to equal&nbsp;1/2. Given the value of the parameter ''β'', what would be the value of the other parameter,&nbsp;''α'', required for the geometric mean to equal&nbsp;1/2?.  The answer is that (for {{nowrap|''β'' > 1}}), the value of ''α'' required tends towards {{nowrap|''β'' + 1/2}} as {{nowrap|''β'' → ∞}}. For example, all these couples have the same geometric mean of&nbsp;1/2: [{{nowrap|1=''β'' = 1, ''α'' = 1.4427}}], [{{nowrap|1=''β'' = 2, ''α'' = 2.46958}}], [{{nowrap|1=''β'' = 3, ''α'' = 3.47943}}], [{{nowrap|1=''β'' = 4, ''α'' = 4.48449}}], [{{nowrap|1=''β'' = 5, ''α'' = 5.48756}}], [{{nowrap|1=''β'' = 10, ''α'' = 10.4938}}], [{{nowrap|1=''β'' = 100, ''α'' = 100.499}}].

The fundamental property of the geometric mean, which can be proven to be false for any other mean, is

:<math>G\left(\frac{X_i}{Y_i}\right) = \frac{G(X_i)}{G(Y_i)}</math>

This makes the geometric mean the only correct mean when averaging ''normalized'' results, that is results that are presented as ratios to reference values.<ref>Philip J. Fleming and John J. Wallace. ''How not to lie with statistics: the correct way to summarize benchmark results''. Communications of the ACM, 29(3):218–221, March 1986.</ref>  This is relevant because the beta distribution is a suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions.  The geometric mean plays a central role in maximum likelihood estimation, see section "Parameter estimation, maximum likelihood." Actually, when performing maximum likelihood estimation, besides the [[geometric mean]] ''G<sub>X</sub>'' based on the random variable X, also another geometric mean appears naturally: the [[geometric mean]] based on the linear transformation ––{{nowrap|(1 − ''X'')}}, the mirror-image of ''X'', denoted by ''G''<sub>(1−''X'')</sub>:

:<math>G_{(1-X)} = e^{\operatorname{E}[\ln(1-X)] } = e^{\psi(\beta) - \psi(\alpha + \beta)}</math>

Along a line {{nowrap|1=''α'' = ''β''}}, the following limits apply:

:<math> \begin{align}
&\lim_{\alpha = \beta \to 0} G_{(1-X)} =0 \\
&\lim_{\alpha = \beta \to \infty} G_{(1-X)} =\tfrac{1}{2}
\end{align}</math>

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

:<math> \begin{align}
\lim_{\beta \to  0} G_{(1-X)} = \lim_{\alpha \to  \infty} G_{(1-X)} = 0\\
\lim_{\alpha\to  0} G_{(1-X)} = \lim_{\beta \to  \infty} G_{(1-X)} = 1
\end{align}</math>

It has the following approximate value:

:<math>G_{(1-X)} \approx \frac{\beta - \frac{1}{2}}{\alpha+\beta-\frac{1}{2}}\text{ if } \alpha, \beta > 1.</math>

Although both ''G''<sub>''X''</sub> and  ''G''<sub>(1−''X'')</sub> are asymmetric, in the case that both shape parameters are equal {{nowrap|1=''α'' = ''β''}}, the geometric means are equal: ''G''<sub>''X''</sub> = ''G''<sub>(1−''X'')</sub>.  This equality follows from the following symmetry displayed between both geometric means:

:<math>G_X (\Beta(\alpha, \beta) )=G_{(1-X)}(\Beta(\beta, \alpha) ). </math>

====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>