Editing Beta distribution (section)

====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).