40  Triangular Distribution

The Triangular distribution requires only three parameters — a minimum, a maximum, and a most-likely value — making it the practical choice when expert knowledge provides only these three bounds. It is widely used in PERT project scheduling, Monte Carlo simulation, and engineering risk assessment.

Formally, the random variate \(X\) defined for the range \(X \in [a, b]\), is said to have a Triangular Distribution (i.e. \(X \sim \text{Triangular}(a, b, c)\)) with minimum \(a\), maximum \(b\), and mode \(c\), where \(a < c < b\).

40.1 Probability Density Function

\[ f(x) = \begin{cases} \dfrac{2(x-a)}{(b-a)(c-a)} & a \leq x \leq c \\[6pt] \dfrac{2(b-x)}{(b-a)(b-c)} & c < x \leq b \end{cases} \]

The figure below shows examples of the Triangular Probability Density Function for different parameter combinations.

Code 
dtriangular <- function(x, a, b, c) {
  ifelse(x < a | x > b, 0,
    ifelse(x <= c,
           2 * (x - a) / ((b - a) * (c - a)),
           2 * (b - x) / ((b - a) * (b - c))))
}

par(mfrow = c(2, 2))

x <- seq(0, 10, length = 500)
plot(x, dtriangular(x, 0, 10, 5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = "a=0, b=10, c=5 (symmetric)")

plot(x, dtriangular(x, 0, 10, 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = "a=0, b=10, c=2 (right-skewed)")

plot(x, dtriangular(x, 0, 10, 8), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = "a=0, b=10, c=8 (left-skewed)")

x2 <- seq(2, 10, length = 500)
plot(x2, dtriangular(x2, 2, 10, 4), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = "a=2, b=10, c=4")

par(mfrow = c(1, 1))
Figure 40.1: Triangular Probability Density Function for various parameter combinations

40.2 Purpose

The Triangular distribution is the most widely used distribution in simulation when only expert-elicited bounds and a most-likely value are available. It requires no knowledge of shape or scale parameters beyond what a domain expert can typically estimate. Common applications include:

  • PERT project scheduling: task duration given optimistic, pessimistic, and most-likely estimates
  • Monte Carlo simulation when only minimum, maximum, and mode are known
  • Engineering design: load, strength, and tolerances with known bounds
  • Financial modeling: revenue and cost estimates from business experts
  • Supply chain: lead times with lower and upper bounds from supplier data

Relation to the discrete setting. The Triangular distribution is the continuous distribution of the sum of two independent Uniform\((0,1)\) variates, restricted to \([0, 2]\) — analogous to how the discrete triangular distribution (sum of two Discrete Uniform variates) arises. In project scheduling, the discrete analog is a finite probability table over integer task durations.

40.3 Distribution Function

\[ F(x) = \begin{cases} \dfrac{(x-a)^2}{(b-a)(c-a)} & a \leq x \leq c \\[6pt] 1 - \dfrac{(b-x)^2}{(b-a)(b-c)} & c < x \leq b \end{cases} \]

The figure below shows the Triangular Distribution Function for \(a = 2\), \(b = 10\), \(c = 4\).

Code 
ptriangular <- function(x, a, b, c) {
  ifelse(x <= a, 0,
    ifelse(x <= c,
           (x - a)^2 / ((b - a) * (c - a)),
           1 - (b - x)^2 / ((b - a) * (b - c))))
}

x <- seq(0, 12, length = 500)
plot(x, ptriangular(x, 2, 10, 4), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Triangular Distribution Function",
     sub = "a = 2, b = 10, c = 4")
Figure 40.2: Triangular Distribution Function (a = 2, b = 10, c = 4)

40.4 Moment Generating Function

\[ M_X(t) = \frac{2\bigl[(b-c)e^{at} - (b-a)e^{ct} + (c-a)e^{bt}\bigr]}{(b-a)(c-a)(b-c)\,t^2} \]

40.5 1st Uncentered Moment

\[ \mu_1' = \frac{a + b + c}{3} \]

40.6 2nd Uncentered Moment

\[ \mu_2' = \frac{a^2 + b^2 + c^2 + ab + ac + bc}{6} \]

40.7 3rd Uncentered Moment

Obtained via the MGF.

40.8 4th Uncentered Moment

Obtained via the MGF.

40.9 2nd Centered Moment

\[ \mu_2 = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18} \]

40.10 3rd Centered Moment

\[ \mu_3 = \frac{(a+b-2c)(2a-b-c)(a-2b+c)}{270} \]

40.11 4th Centered Moment

\[ \mu_4 = \frac{12}{5}\mu_2^2 \]

Note: since \(g_2 = 12/5\), we have \(\mu_4 = (12/5)\mu_2^2\).

40.12 Expected Value

\[ \text{E}(X) = \frac{a + b + c}{3} \]

40.13 Variance

\[ \text{V}(X) = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18} \]

40.14 Median

\[ \text{Med}(X) = \begin{cases} a + \sqrt{(b-a)(c-a)/2} & \text{if } c \geq (a+b)/2 \\[4pt] b - \sqrt{(b-a)(b-c)/2} & \text{if } c < (a+b)/2 \end{cases} \]

40.15 Mode

\[ \text{Mo}(X) = c \]

40.16 Coefficient of Skewness

\[ g_1 = \frac{\sqrt{2}(a+b-2c)(2a-b-c)(a-2b+c)}{5\,(a^2+b^2+c^2-ab-ac-bc)^{3/2}} \]

For the symmetric case \(c = (a+b)/2\): \(g_1 = 0\).

40.17 Coefficient of Kurtosis

\[ g_2 = \frac{12}{5} = 2.4 \]

This is a remarkable fixed constant — the kurtosis is always 2.4, regardless of \(a\), \(b\), and \(c\). The Triangular distribution always has lighter tails than the Normal (\(g_2 = 3\)).

40.18 Parameter Estimation

In practice, parameters are estimated from expert elicitation or sample extremes:

\[ \hat a = x_{(1)}, \quad \hat b = x_{(n)}, \quad \hat c = \text{sample mode estimate} \]

For simulation purposes, the parameters are typically given directly by domain experts rather than estimated from data.

# Example: verify mean and variance for Triangular(2, 10, 4)
a <- 2; b <- 10; c <- 4

mean_tri <- (a + b + c) / 3
var_tri  <- (a^2 + b^2 + c^2 - a*b - a*c - b*c) / 18

cat("Mean:", round(mean_tri, 4), "\n")
cat("Variance:", round(var_tri, 4), "\n")
cat("SD:", round(sqrt(var_tri), 4), "\n")
cat("Kurtosis g2:", 12/5, "(always 2.4)\n")
Mean: 5.3333 
Variance: 2.8889 
SD: 1.6997 
Kurtosis g2: 2.4 (always 2.4)

40.19 R Module

40.19.1 RFC

The Triangular Distribution module is available in RFC under the menu “Distributions / Triangular Distribution”.

40.19.3 R Code

The following code demonstrates Triangular probability calculations:

a <- 2; b <- 10; c <- 4

ptriangular <- function(x, a, b, c) {
  ifelse(x <= a, 0,
    ifelse(x <= c,
           (x - a)^2 / ((b - a) * (c - a)),
           1 - (b - x)^2 / ((b - a) * (b - c))))
}

# P(task <= 5 days)
ptriangular(5, a, b, c)

# P(task <= mean)
ptriangular((a + b + c)/3, a, b, c)

# Mean
cat("Mean:", (a + b + c) / 3, "\n")
cat("Mode:", c, "\n")
[1] 0.4791667
[1] 0.5462963
Mean: 5.333333 
Mode: 4

40.20 Example

A project task has a minimum duration of \(a = 2\) days, maximum \(b = 10\) days, and most-likely duration \(c = 4\) days: \(X \sim \text{Triangular}(2, 10, 4)\). The expected duration is \((2+10+4)/3 \approx 5.33\) days.

a <- 2; b <- 10; c <- 4

ptriangular <- function(x, a, b, c) {
  ifelse(x <= a, 0,
    ifelse(x <= c,
           (x - a)^2 / ((b - a) * (c - a)),
           1 - (b - x)^2 / ((b - a) * (b - c))))
}

# Expected duration
cat("Expected duration (days):", (a + b + c) / 3, "\n")

# P(task done within 5 days)
cat("P(done within 5 days):", round(ptriangular(5, a, b, c), 4), "\n")

# P(task takes more than 8 days)
cat("P(takes > 8 days):", round(1 - ptriangular(8, a, b, c), 4), "\n")
Expected duration (days): 5.333333 
P(done within 5 days): 0.4792 
P(takes > 8 days): 0.0833
Interactive Shiny app (click to load).
Open in new tab

40.21 Random Number Generator

Triangular random variates are generated via the inverse-CDF method. Let \(p_c = (c-a)/(b-a)\):

\[ X = \begin{cases} a + \sqrt{U(b-a)(c-a)} & U \leq p_c \\[4pt] b - \sqrt{(1-U)(b-a)(b-c)} & U > p_c \end{cases} \]

set.seed(123)
n <- 10000; a <- 2; b <- 10; c <- 4

pc <- (c - a) / (b - a)
u  <- runif(n)
x_sim <- ifelse(u <= pc,
                a + sqrt(u * (b - a) * (c - a)),
                b - sqrt((1 - u) * (b - a) * (b - c)))

cat("Simulated mean:", round(mean(x_sim), 4), "\n")
cat("Theoretical mean:", (a + b + c) / 3, "\n")
cat("Simulated var:", round(var(x_sim), 4), "\n")
cat("Theoretical var:",
    (a^2 + b^2 + c^2 - a*b - a*c - b*c) / 18, "\n")
Simulated mean: 5.3143 
Theoretical mean: 5.333333 
Simulated var: 2.8332 
Theoretical var: 2.888889
Interactive Shiny app (click to load).
Open in new tab

40.22 Property 1: Fixed Kurtosis

The kurtosis \(g_2 = 12/5 = 2.4\) is always fixed, regardless of \(a\), \(b\), and \(c\). This means the Triangular distribution is always platykurtic (lighter-tailed than the Normal), which follows from the bounded support.

40.23 Property 2: Symmetric Special Case

When \(c = (a+b)/2\) the distribution is symmetric (\(g_1 = 0\)) and resembles a tent function. The Uniform distribution is not the limit of this case — the symmetric Triangular distribution retains its triangular density.

40.24 Property 3: Sum of Two Uniforms

The Triangular\((0, 2, 1)\) distribution is the distribution of the sum of two independent Uniform\((0,1)\) random variables. More generally, Triangular\((a, b, (a+b)/2)\) arises as a scaled and shifted sum of two Uniforms.