pert
The (Modified) PERT Distribution
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.
 Keywords
 distribution
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].
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
$$\alpha_{1}=\frac{(\mumin)(2\times modeminmax)}{(mode\mu)(maxmin)}$$
$$\alpha_{2}=\frac{\alpha_{1}\times (max\mu)}{mumin}$$
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.
Value

dpert gives the density, ppert gives the distribution
function, qpert gives the quantile function, and rpert
generates random deviates.
References
Vose D. Risk Analysis  A Quantitative Guide (2nd and 3rd editions, John Wiley and Sons, 2000, 2008).
See Also
Examples
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)