mc2d (version 0.1-17)

# pert: The (Modified) PERT Distribution

## Description

Density, distribution function, quantile function and random generation for the PERT (aka Beta PERT) distribution with minimum equals to min, mode equals to mode and maximum equals to max.

## Usage

dpert(x, min=-1, mode=0, max=1, shape=4, log=FALSE)
ppert(q, min=-1, mode=0, max=1, shape=4, lower.tail=TRUE, log.p=FALSE)
qpert(p, min=-1, mode=0, max=1, shape=4, lower.tail=TRUE, log.p=FALSE)
rpert(n, min=-1, mode=0, max=1, shape=4)

## Arguments

x,q
Vector of quantiles.
p
Vector of probabilities.
n
Number of observations. If length(n) $>$ 1, the length is taken to be the number required.
min
Vector of minima.
mode
Vector of modes.
max
Vector of maxima.
shape
Vector of scaling parameters. Default value: 4.
log, log.p
Logical; if TRUE, probabilities p are given as log(p).
lower.tail
Logical; if TRUE (default), probabilities are P[X <= x]<="" span="">, otherwise, P[X > x].

## Value

dpert gives the density, ppert gives the distribution function, qpert gives the quantile function, and rpert generates random deviates.

## Details

The PERT distribution is a beta distribution extended to the domain [min, max] with mean $$\mu=\frac{min+shape\times mode+max}{shape+2}$$

The underlying beta distribution is specified by $shape1$ and $shape2$ defined as

$$\alpha_{1}=\frac{(\mu-min)(2\times mode-min-max)}{(mode-\mu)(max-min)}$$

$$\alpha_{2}=\frac{\alpha_{1}\times (max-\mu)}{mu-min}$$

If $mu=mode$, $shape1$ is set to $1+shape/2$.

David Vose (See reference) proposed a modified PERT distribution with a shape parameter different from 4.

The PERT distribution is frequently used (with the triangular distribution) to translate expert estimates of the min, max and mode of a random variable in a smooth parametric distribution.

## References

Vose D. Risk Analysis - A Quantitative Guide (2nd and 3rd editions, John Wiley and Sons, 2000, 2008).

Beta

## Examples

Run this code
curve(dpert(x, min=3, mode=5, max=10, shape=6), from = 2, to = 11, lty=3)
curve(dpert(x, min=3, mode=5, max=10), from = 2, to = 11, add=TRUE)
curve(dpert(x, min=3, mode=5, max=10, shape=2), from = 2, to = 11, add=TRUE, lty=2)
legend(x = 8, y = 2, c("Default", "shape:2", "shape:6"), lty=1:3)



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