extraDistr (version 1.10.0)

Triangular: Triangular distribution

Description

Density, distribution function, quantile function and random generation for the triangular distribution.

Usage

dtriang(x, a = -1, b = 1, c = (a + b)/2, log = FALSE)
ptriang(q, a = -1, b = 1, c = (a + b)/2, lower.tail = TRUE, log.p = FALSE)
qtriang(p, a = -1, b = 1, c = (a + b)/2, lower.tail = TRUE, log.p = FALSE)
rtriang(n, a = -1, b = 1, c = (a + b)/2)

Arguments

x, q

vector of quantiles.

a, b, c

minimum, maximum and mode of the distribution.

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are $P[X\le x]$ 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

Probability density function $$f(x)=\{\begin{array}{ll}\frac{2(x-a)}{(b-a)(c-a)}& x<c\\ \frac{2}{b-a}& x=c\\ \frac{2(b-x)}{(b-a)(b-c)}& x>c\end{array}$$

Cumulative distribution function $$F(x)=\{\begin{array}{ll}\frac{(x-a{)}^2}{(b-a)(c-a)}& x\le c\\ 1-\frac{(b-x{)}^2}{(b-a)(b-c)}& x>c\end{array}$$

Quantile function $$F^{-1}(p)=\{\begin{array}{ll}a+\sqrt{p\times (b-a)(c-a)}& p\le \frac{c-a}{b-a}\\ b-\sqrt{(1-p)(b-a)(b-c)}& p>\frac{c-a}{b-a}\end{array}$$

For random generation MINMAX method described by Stein and Keblis (2009) is used.

References

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Stein, W. E., & Keblis, M. F. (2009). A new method to simulate the triangular distribution. Mathematical and computer modelling, 49(5), 1143-1147.

Examples

x <- rtriang(1e5, 5, 7, 6)
hist(x, 100, freq = FALSE)
curve(dtriang(x, 5, 7, 6), 3, 10, n = 500, col = "red", add = TRUE)
hist(ptriang(x, 5, 7, 6))
plot(ecdf(x))
curve(ptriang(x, 5, 7, 6), 3, 10, n = 500, col = "red", lwd = 2, add = TRUE)