Beta Rectangular
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0}$shape (real) ${\displaystyle \beta >0}$shape (real) mixture parameter
Support	${\displaystyle x\in (a,b)}$
PDF
CDF	where${\displaystyle z=(x-a)/(b-a)}$
Mean	${\displaystyle a+(b-a)\left(\frac{\theta \alpha }{\alpha +\beta }+\frac{1-\theta }{2}\right)}$
Variance	${\displaystyle (b-a{)}^2\left(\frac{\theta \alpha (\alpha +1)}{k(k+1)}+\frac{1-\theta }{3}-\frac{(k+\theta (\alpha -\beta ){)}^2}{4k^2}\right)}$ where${\displaystyle k=\alpha +\beta }$

Inprobability theory andstatistics, thebeta rectangular distribution is aprobability distribution that is a finitemixture distribution of thebeta distribution and thecontinuous uniform distribution. The support is of the distribution is indicated by the parametersa andb, which are the minimum and maximum values respectively. The distribution provides an alternative to the beta distribution such that it allows more density to be placed at the extremes of the bounded interval of support.^[1] Thus it is a bounded distribution that allows foroutliers to have a greater chance of occurring than does the beta distribution.

If parameters of the beta distribution areα andβ, and if the mixture parameter isθ, then the beta rectangular distribution hasprobability density function^{[citation needed]}

where${\displaystyle \mathrm{\Gamma }(\cdot )}$ is thegamma function.

Thecumulative distribution function is^{[citation needed]}

${\displaystyle F(x|\alpha ,\beta ,\theta )=\theta I_z(\alpha ,\beta )+\frac{(1-\theta )(x-a)}{b-a}\mathrm{f}\mathrm{o}\mathrm{r}a\le x\le b,}$

where${\displaystyle z={\displaystyle \frac{x-a}{b-a}}}$ and${\displaystyle I_z(\alpha ,\beta )}$ is theregularized incomplete beta function.

ThePERT distribution variation of thebeta distribution is frequently used inPERT,critical path method (CPM) and otherproject management methodologies to characterize the distribution of an activity's time to completion.^[2]

In PERT, restrictions on the PERT distribution parameters lead to shorthand computations for the mean and standard deviation of the beta distribution:

wherea is the minimum,b is the maximum, andm is the mode or most likely value. However, the variance is seen to be a constant conditional on the range. As a result, there is no scope for expressing differing levels of uncertainty that the project manager might have about the activity time.

Eliciting the beta rectangular's certainty parameterθ allows the project manager to incorporate the rectangular distribution and increase uncertainty by specifyingθ is less than 1. The above expectation formula then becomes

${\displaystyle E(x)=\frac{\theta (a+4m+b)+3(1-\theta )(a+b)}{6}.}$

If the project manager assumes the beta distribution is symmetric under the standard PERT conditions then the variance is

${\displaystyle \mathrm{Var}(x)=\frac{(b-a{)}^2(3-2\theta )}{36},}$

while for the asymmetric case it is

${\displaystyle \mathrm{Var}(x)=\frac{(b-a{)}^2(3-2{\theta }^2)}{36}.}$

The variance can now be increased when uncertainty is larger. However, the beta distribution may still apply depending on the project manager's judgment.

The beta rectangular has been compared to the uniform-two sided power distribution and the uniform-generalized biparabolic distribution in the context of project management. The beta rectangular exhibited larger variance and smaller kurtosis by comparison.^[3]

The beta rectangular distribution has been compared to the elevated two-sided power distribution in fitting U.S. income data.^[4] The 5-parameter elevated two-sided power distribution was found to have a better fit for some subpopulations, while the 3-parameter beta rectangular was found to have a better fit for other subpopulations.

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• Compound probability distributions

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