The Triangular Distribution

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

dtri(x, min = 0, max = 1, mode = 1/2)
  ptri(q, min = 0, max = 1, mode = 1/2)
  qtri(p, min = 0, max = 1, mode = 1/2)
  rtri(n, min = 0, max = 1, mode = 1/2)

Arguments

x

vector of quantiles. Missing values (NAs) are allowed.

q

vector of quantiles. Missing values (NAs) are allowed.

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.

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.

Details

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: $$E(X) = \frac{a + b + c}{3}$$ $$Var(X) = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18}$$

Value

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

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

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=\((\beta-\alpha)/2\).

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

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