Triangular distribution

When you know the minimum, maximum, and most-likely value of something but nothing else, the triangular distribution is the natural stand-in. It shows up in PERT schedules, Monte Carlo risk models, and anywhere a practitioner has to guess at uncertainty with three anchors.

f (x; a, b, c) = ⎩ ⎨ ⎧ \frac{2 ( x - a )}{( b - a ) ( c - a )} \frac{2 ( b - x )}{( b - a ) ( b - c )} 0 a \leq x \leq c c < x \leq b otherwise

a (min) 0.0 c (mode) 3.0 b (max) 10.0 n (samples) 1000

Piecewise-linear density on [a, b] peaking at c. Mean is the average of the three vertices

(a + b + c) /3

= 4.33.

What to notice

Three knobs, piecewise linear. The density is a straight line up from a to the mode c and a straight line down to b. The peak height is always $2/ (b - a)$ , independent of where the mode sits.
Mode ≠ mean. Drag the mode toward one side and the mean stays at the average of the three vertices:

E [X] = (a + b + c) /3

Symmetric case. When c sits exactly at the midpoint, the distribution is symmetric and its variance simplifies to $(b - a)^{2} /24$ — the same as two uniforms added together.

Why it matters

It’s what you reach for when you can’t justify a Normal but have to sample something. In project-estimation tradition the mean is often approximated as $(a + 4 c + b) /6$ , a PERT-style weighted average that sits between the triangular and a Beta fitted to the same three anchors.