Triangular: The triangular distribution.

Description

Density, distribution function, quantile function and random generation for the triangular distribution with range 'min' to 'max' and mode equal to 'mode'.

Usage

dtri(x, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, 
		log = FALSE)
		
ptri(q, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, 
		lower.tail = TRUE, log.p = FALSE)
		
qtri(p, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2, 
		lower.tail = TRUE, log.p = FALSE)
		
rtri(n, min=-sqrt(6), max=sqrt(6), mode = (min + max)/2)

Arguments

x, q

Vector of quantiles.

Number of observations. If 'length(n) > 1', the length is taken to be the number required.

min

Vector of lower limits of distribution.

max

Vector of upper limits of distribution.

mode

Vector of modes

lower.tail

logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x].

log, log.p

logical; if TRUE, probabilities p are given as log(p).

Value

A vector of densities, probabilities, quantiles or random deviates. dtri gives the density, ptri gives the distribution function, qtri gives the quantile function, and rtri generates random deviates.

Details

The triangular distribution has density $$f(x)=2*(x-min) / ((max-min)*(mode-min)) (min < x <= mode)$$ $$f(x)=2*(max-x) / ((max-min)*(max-mode)) (mode < x < max)$$ and 0 elsewhere.

The mean is $$\frac{1}{3}(min + mode + max)$$ and the variance is $$\frac{1}{18}(min^2 + mode^2 + max^2 - min*mode - min*max - mode*max)$$

The default values of min, max and mode give a distribution with mean 0 and unit variance.

If min>max, min amd max will be silently interchanged. If mode is not within [min, max], the functions return NA, with a warning.

rtri calls runif(n, 0, 1) to generate probabilities which are passed to qtri.

Examples

Run this code

# NOT RUN {
     require(graphics)
     x<-seq(-3,3,0.02)

     par(mfrow=c(2,1))
     plot(x, dtri(x), type="l", main="Density")
     plot(x, ptri(x), type="l", main="p(X<x)")

     u <- rtri(5000)

     var(rtri(10000,-1,1))  # ~ = 1/6 = 0.167

# }