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Triangular distribution
You will learn: Compute areas and cumulative probabilities when a bounded density has a specified mode.
Start with: Uniform distribution
When you know the minimum, maximum, and most-likely value of something but nothing else, a triangular distribution is one simple modeling choice. Three elicited values do not uniquely determine a distribution. The important question is what changes when those anchors or the density between them are uncertain.
Three anchors still leave a modeling choice
| Model | Mean | P(X≤5) |
|---|---|---|
| Triangular, mode 3 | 4.33333 | 0.642857 |
| Uniform | 5.00000 | 0.500000 |
| Beta-PERT, strength 4 | 3.66667 | 0.760011 |
The beta alternative maps Beta(2.200,3.800) to [a,b], using an explicit strength convention of 4. It shares the elicited mode, but has a different shape. The uniform comparison uses only the bounds and has no unique mode. These are assumptions, not three fitted conclusions from observed data.
Move the elicited mode and inspect the consequence
| Mode | Mean | P(X≤5) |
|---|---|---|
| 0.500 | 3.50000 | 0.736842 |
| 3.000 | 4.33333 | 0.642857 |
| 5.500 | 5.16667 | 0.454545 |
A mode change Δ moves the mean by Δ/3; the event probability need not change linearly. The entered triangular variance is 4.38889. Bounds describe assumed possible values, not automatic confidence limits. An outcome outside them has probability zero in all three models.
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 , 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:
- Symmetric case. When c sits exactly at the midpoint, the distribution is symmetric and its variance simplifies to — the variance of a plus the sum of two independent Uniform(0, (b−a)/2) variables.
Why it matters
A simple bounded model can make assumptions inspectable, but simplicity is not evidence of fit. In project-estimation tradition the mean is often approximated as , the usual beta-PERT mean under its additional shape convention. It is not the triangular mean and is not determined by the three anchors alone.
The displayed two-branch density assumes a < c < b. At c=a or c=b, use the corresponding one-sided triangular limit; do not evaluate a zero-width branch by dividing by zero.
Compute an event area
With a=0, b=10, c=3, the probability up to the mode is the area of the left triangle: (3−0)/(10−0)=0.3. Above the mode, subtract the right-hand triangle from one. At x=5, P(X≤5)=1−(10−5)²/((10−0)(10−3))=9/14≈0.642857. A uniform distribution on the same bounds instead gives 0.5.
The mean is 13/3≈4.333333. If only c changes from 3 to 6, the mean rises by one to 16/3, but P(X≤5) falls to 25/60≈0.416667. The cutoff has moved from one branch of the density to the other, so a linear rule for the event probability would fail.
Make the alternative shape convention explicit
The comparison uses a beta-PERT convention with strength 4. On the unit interval its shapes are α=1+4(c−a)/(b−a) and β=1+4(b−c)/(b−a). Scaling to [a,b] produces mean (a+4c+b)/6. When the mode is interior, the beta mode formula returns the elicited c; at a boundary it gives the corresponding monotone density.
That convention is an additional assumption. It does not follow from the three anchors. A uniform density has no unique mode, and other beta strengths can produce other event probabilities. The table holds the support and cutoff fixed so the consequence of changing the shape is visible.
Make a prediction
If a reported maximum is merely the largest observation so far, is it safe to use it as b?
Explore the answer
Not without a support assumption. A triangular model assigns zero probability above b. A sample maximum does not establish an absolute upper limit. Reconsider the bound or the family when observations can exceed it.
References
NIST’s triangular density reference gives the two branches on general bounds. The areas and beta-PERT convention above are derived from the displayed densities and shape formulas. Use beta and uniform to compare their assumptions.