Triangular Distribution

The triangular distribution is a continuous distribution with a lower bound, an upper bound, and a mode, producing a piecewise linear, triangular-shaped density function. It is commonly used in risk assessment and simulation when only the minimum, maximum, and most likely value of a quantity are known.

Usage

dtri(x, min = 0, max = 1, mode = 1/2, log = FALSE)

ptri(q, min = 0, max = 1, mode = 1/2, lower.tail = TRUE, log.p = FALSE)

qtri(p, min = 0, max = 1, mode = 1/2, lower.tail = TRUE, log.p = FALSE)

rtri(n, min = 0, max = 1, mode = 1/2)

Arguments

x: vector of quantiles. Missing values (NAs) are allowed.
min: vector of minimum values of the distribution of the random variable. The default value is min=0.
max: vector of maximum values of the random variable. The default value is max=1.
mode: vector of modes of the random variable. The default value is mode=1/2. The parameters must satisfy $min < mode < max$.
log, log.p: logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.
q: vector of quantiles. Missing values (NAs) are allowed.
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p: vector of probabilities between 0 and 1. Missing values (NAs) are allowed.
n: sample size. If length(n) is larger than 1, then length(n) random values are returned.

Value

dtri() gives the density, ptri() gives the distribution function, qtri() gives the quantile function, and rtri() generates random deviates.

Details

Density, distribution function, quantile function, and random generation for the triangular distribution with parameters min, max, and mode.

Let $X$ be a triangular random variable with parameters min=$a$, max=$b$, and mode=$c$.

Probability Density and Cumulative Distribution Function
The density function of $X$ is given by:

$f(x; a, b, c) =$	$\frac{2(x-a)}{(b-a)(c-a)}$	for $a \le x \le c$
	$\frac{2(b-x)}{(b-a)(b-c)}$	for $c \le x \le b$

where $a < c < b$.

The cumulative distribution function of $X$ is given by:

$F(x; a, b, c) =$	$\frac{(x-a)^2}{(b-a)(c-a)}$	for $a \le x \le c$
	$1 - \frac{(b-x)^2}{(b-a)(b-c)}$	for $c \le x \le b$

where $a < c < b$.

Quantiles
The $p^th$ quantile of $X$ is given by:

$x_p =$	$a + \sqrt{(b-a)(c-a)p}$	for $0 \le p \le F(c)$
	$b - \sqrt{(b-a)(b-c)(1-p}$	for $F(c) \le p \le 1$

where $0 \le p \le 1$.

Random Numbers
Random numbers are generated using the inverse transformation method: $$x = F^{-1}(u)$$ where $u$ is a random deviate from a uniform $[0, 1]$ distribution.

Mean and Variance
The mean and variance of $X$ are given by: $$E(X) = \frac{a + b + c}{3}$$ $$Var(X) = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18}$$

The triangular distribution is so named because of the shape of its probability density function. The average of two independent identically distributed uniform random variables with parameters min=$\alpha$ and max=$\beta$ has a triangular distribution with parameters min=$\alpha$, max=$\beta$, and mode=$(\alpha+\beta)/2$.

The triangular distribution is sometimes used as an input distribution in probability risk assessment.

Note

Based on code by Steven P. Millard previously published in the EnvStats package, adapted to conform to package standards.

References

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.

Examples


# Density of a triangular distribution with parameters 
# min=10, max=15, and mode=12, evaluated at 12, 13 and 14: 

dtri(12:14, 10, 15, 12) 
#> [1] 0.4000000 0.2666667 0.1333333
## [1] 0.4000000 0.2666667 0.1333333

# The cdf of a triangular distribution with parameters 
# min=2, max=7, and mode=5, evaluated at 3, 4, and 5: 

ptri(3:5, 2, 7, 5) 
#> [1] 0.06666667 0.26666667 0.60000000
## [1] 0.06666667 0.26666667 0.60000000

# The 25'th percentile of a triangular distribution with parameters 
# min=1, max=4, and mode=3: 

qtri(0.25, 1, 4, 3) 
#> [1] 2.224745
## [1] 2.224745

# A random sample of 4 numbers from a triangular distribution with 
# parameters min=3 , max=20, and mode=12. 
# (Note: the call to set.seed simply allows you to reproduce this example.)

set.seed(10) 
rtri(4, 3, 20, 12) 
#> [1] 11.811593  9.850955 11.081885 13.539496
## [1] 11.811593  9.850955 11.081885 13.539496

\(f(x; a, b, c) =\)	\(\frac{2(x-a)}{(b-a)(c-a)}\)	for \(a \le x \le c\)
	\(\frac{2(b-x)}{(b-a)(b-c)}\)	for \(c \le x \le b\)

\(F(x; a, b, c) =\)	\(\frac{(x-a)^2}{(b-a)(c-a)}\)	for \(a \le x \le c\)
	\(1 - \frac{(b-x)^2}{(b-a)(b-c)}\)	for \(c \le x \le b\)

\(x_p =\)	\(a + \sqrt{(b-a)(c-a)p}\)	for \(0 \le p \le F(c)\)
	\(b - \sqrt{(b-a)(b-c)(1-p}\)	for \(F(c) \le p \le 1\)