The triangular distribution

What it models

• The triangular distribution is used when you know three things about a quantity:
  • a minimum possible value, $a$
  • a maximum possible value, $b$
  • the most likely value — the mode — $c$, somewhere between them.
• It is the natural model when an expert estimate is all that is available: "it will take at least 3 days, at most 12, and most likely 5". No data, three judgements, one distribution.
• Typical contexts: project completion times, cost estimates, crop yields, the time to repair a fault.

The shape

• The density is a triangle: zero at $a$, rising to a peak at $c$, falling back to zero at $b$.
• Because the total area must be 1, and a triangle's area is $\tfrac{1}{2} \times \text{base} \times \text{height}$:

$$\tfrac{1}{2}(b-a) \times h = 1 \qquad \Rightarrow \qquad h = \frac{2}{b-a}$$

• The peak height depends only on the range, not on where the mode sits. Moving the mode changes the shape but not the height.

Finding probabilities — always by area

• There is no calculator function for this. Every triangular probability is a geometric area calculation, and the method is:

  • Sketch the triangle with $a$, $c$, $b$ and the height $\frac{2}{b-a}$ marked.
  • Shade the region you want.
  • Decide whether it is easier to find that area directly or to find the complement and subtract.
  • Use similar triangles to get the height of the density at the boundary of the region.

• The key technique is similar triangles. For a point $x$ on the left slope (between $a$ and $c$), the density height there is:

$$h(x) = \frac{2}{b-a} \times \frac{x - a}{c - a}$$

• and the area to the left of $x$ is the small triangle:

$$P(X < x) = \tfrac{1}{2}(x - a) \times h(x) = \frac{(x-a)^2}{(b-a)(c-a)}$$

• Similarly, for $x$ on the right slope (between $c$ and $b$), the area to the right is a small triangle:

$$P(X > x) = \frac{(b-x)^2}{(b-a)(b-c)}$$

• You do not have to memorise these. If you sketch the triangle and use similar triangles, you will derive them each time — and the specification credits the working.

A check worth doing every time

• The area to the left of the mode is:

$$P(X < c) = \frac{c - a}{b - a}$$

• This tells you instantly which side of the triangle any given value falls on, and whether your answer should be above or below that figure.

Mean and standard deviation

$$\mu = \frac{a + b + c}{3}$$

• The mean is the average of the three parameters — a pleasingly simple result. Note that it is not the mode, unless the triangle is symmetric.
• The mean sits between the mode and the midpoint of the range, pulled towards the longer tail. If $c$ is left of centre, the distribution is right-skewed and $\mu > c$.

Choosing the parameters

• Exam questions frequently ask you to suggest and justify parameters for a triangular model of a described situation. A good answer:
  • sets $a$ and $b$ at values that are genuinely possible extremes, not at typical values
  • sets $c$ at the most commonly occurring value described
  • justifies each from the context, not from arithmetic convenience
  • notes that the model assumes the density falls linearly from the mode to each extreme, which is an approximation.