Editing Beta distribution (section)

===Project management: task cost and schedule modeling===
The beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the beta distribution&nbsp;— along with the [[triangular distribution]]&nbsp;— is used extensively in [[PERT]], [[critical path method]] (CPM), Joint Cost Schedule Modeling (JCSM) and other [[project management]]/control systems to describe the time to completion and the cost of a task. In project management, shorthand computations are widely used to estimate the [[mean]] and [[standard deviation]] of the beta distribution:<ref name=Malcolm>{{cite journal |last1=Malcolm |first1=D. G. |last2=Roseboom |first2=J. H. |last3=Clark |first3=C. E. |last4=Fazar |first4=W. |title=Application of a Technique for Research and Development Program Evaluation |journal=Operations Research |date=September–October 1958 |volume=7 |issue=5 |pages=646–669 |doi=10.1287/opre.7.5.646 |issn=0030-364X}}</ref>

:<math> \begin{align}
  \mu(X) & = \frac{a + 4b + c}{6} \\[8pt]
  \sigma(X) & = \frac{c-a}{6}
\end{align}</math>

where ''a'' is the minimum, ''c'' is the maximum, and ''b'' is the most likely value (the [[Mode (statistics)|mode]] for ''α'' > 1 and ''β'' > 1).

The above estimate for the [[mean]] <math>\mu(X)= \frac{a + 4b + c}{6}</math> is known as the [[PERT]] [[three-point estimation]] and it is exact for either of the following values of ''β'' (for arbitrary α within these ranges):

:''β'' = ''α'' > 1 (symmetric case) with [[standard deviation]] <math>\sigma(X) = \frac{c-a}{2 \sqrt {1+2\alpha}}</math>, [[skewness]] = 0, and [[excess kurtosis]] = <math> \frac{-6}{3+2 \alpha}</math>
[[File:Beta_Distribution_beta=alpha_from_1.05_to_4.95.svg|thumb]]

or

:''β'' = 6 − ''α'' for 5 > ''α'' > 1 (skewed case) with [[standard deviation]]

:<math>\sigma(X) = \frac{(c-a)\sqrt{\alpha(6-\alpha)}}{6 \sqrt 7},</math>

[[skewness]]<math>{}=\frac{(3-\alpha) \sqrt 7}{2\sqrt{\alpha(6-\alpha)}}</math>, and [[excess kurtosis]]<math>{}=\frac{21}{\alpha (6- \alpha)} - 3</math>

[[File:Beta Distribution beta=6-alpha from 1.05 to 4.95.svg|thumb]]

The above estimate for the [[standard deviation]] ''σ''(''X'') = (''c'' − ''a'')/6 is exact for either of the following values of ''α'' and ''β'':

:''α'' = ''β'' = 4 (symmetric) with [[skewness]] = 0, and [[excess kurtosis]] = −6/11.
:''β'' = 6 − ''α'' and <math>\alpha = 3 - \sqrt2</math> (right-tailed, positive skew) with [[skewness]]<math>{}=\frac{1}{\sqrt 2}</math>, and [[excess kurtosis]] = 0
:''β'' = 6 − ''α'' and <math>\alpha = 3 + \sqrt2</math> (left-tailed, negative skew) with [[skewness]]<math>{}= \frac{-1}{\sqrt 2}</math>, and [[excess kurtosis]] = 0

[[File:Beta Distribution for conjugate alpha beta.svg|thumb]]

Otherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.<ref>Keefer, Donald L. and Verdini, William A. (1993). Better Estimation of PERT Activity Time Parameters. Management Science 39(9), p. 1086&ndash;1091.</ref><ref>Keefer, Donald L. and Bodily, Samuel E. (1983). Three-point Approximations for Continuous Random variables. Management Science 29(5), p. 595&ndash;609.</ref><ref>{{Cite web|url=https://www.nps.edu/web/drmi/|title=Defense Resource Management Institute - Naval Postgraduate School|website=www.nps.edu}}</ref>