pert: The (Modified) PERT Distribution

In mc2d: Tools for Two-Dimensional Monte-Carlo Simulations

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’ (or, alternatively, mean equals to ‘⁠mean⁠’) and maximum equals to ‘⁠max⁠’.

Usage

dpert(x, min = -1, mode = 0, max = 1, shape = 4, log = FALSE, mean = 0)

ppert(
  q,
  min = -1,
  mode = 0,
  max = 1,
  shape = 4,
  lower.tail = TRUE,
  log.p = FALSE,
  mean = 0
)

qpert(
  p,
  min = -1,
  mode = 0,
  max = 1,
  shape = 4,
  lower.tail = TRUE,
  log.p = FALSE,
  mean = 0
)

rpert(n, min = -1, mode = 0, max = 1, shape = 4, mean = 0)

Arguments

x, q	Vector of quantiles.
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)⁠’.
mean	Vector of means, can be specified in place of ‘⁠mode⁠’ as an alternative parametrization.
lower.tail	Logical; if ‘⁠TRUE⁠’ (default), probabilities are ‘⁠P[X <= x]⁠’, otherwise, ‘⁠P[X > x]⁠’
p	Vector of probabilities
n	Number of observations. If length(n) > 1, the length is taken to be the number required.

Details

The PERT distribution is a Beta distribution extended to the domain ‘⁠[min, max]⁠’ with mean

mean=\frac{min+shape\times mode+max}{shape+2}

The underlying beta distribution is specified by \alpha_{1} and \alpha_{2} defined as

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

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

‘⁠mode⁠’ or ‘⁠mean⁠’ can be specified, but not both. Note: ‘⁠mean⁠’ is the last parameter for back-compatibility. A warning will be provided if some combinations of ‘⁠min⁠’, ‘⁠mean⁠’ and ‘⁠max⁠’ leads to impossible mode.

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.

Value

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

Author(s)

Regis Pouillot and Matthew Wiener

References

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

See Also

Beta

Examples

curve(dpert(x,min=3,mode=5,max=10,shape=6), from = 2, to = 11, lty=3,ylab="density")
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 = .30, c("Default: 4","shape: 2","shape: 6"), lty=1:3)
## Alternatie parametrization using mean
curve(dpert(x,min=3,mean=5,max=10), from = 2, to = 11, lty=2 ,ylab="density")
curve(dpert(x,min=3,mode=5,max=10), from = 2, to = 11, add=TRUE)
legend(x = 8, y = .30, c("mode: 5","mean: 5"), lty=1:2)

mc2d documentation built on July 26, 2023, 6:07 p.m.