Library / Probability distributions / PRD-05
Probability distributions
Triangular estimate: probability density
Height of the triangular density at a test value x, for a quantity with a lower bound a, a most likely value m and an upper bound b.
- Ref
- PRD-05
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/day]
The triangular density rises in a straight line from zero at a to a peak of 2/(b − a) at the most likely value m, then falls in a straight line to zero at b. The two branches of the formula cover the two sides of the peak. Outside [a, b] the density is zero. The model needs a < m < b.
Height of the triangular density at a test value x, for a quantity with a lower bound a, a most likely value m and an upper bound b. It rises in a straight line to a peak at m, then falls to zero at b, and is zero outside [a, b]. The first form applies up to m and the second above it.
Calculator
- Peak density, 2/(b − a)0.25 1/day
- Range, b − a8 days
- Mean of the distribution4.333 days
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 4 days the density is 0.2083 per day, against a peak of 0.25 per day at the most likely value of 3 days.
- A wider range between a and b lowers the peak, because the same unit area is spread over more days.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/day) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Upper bound b | 3.5 to 20 days | 0 | 0.2679 | 0.2679 | +10% | −7.77 % |
| x Test value x | 0 to 12 days | 0 | 0.2417 | 0.2417 | +10% | −8 % |
| m Most likely value m | 1.5 to 8.5 days | 0.1 | 0.2446 | 0.1446 | +10% | +5.26 % |
| a Lower bound a | −5 to 2.5 days | 0.119 | 0.2564 | 0.1374 | +0.15 days | +1.91 % |
Sweep of Test value x
| Test value x (days) | Probability density (1/day) |
|---|---|
| 0 | 0 |
| 1.5 | 0.0625 |
| 3 | 0.25 |
| 4.5 | 0.1875 |
| 6 | 0.125 |
| 7.5 | 0.0625 |
| 9 | 0 |
| 10.5 | 0 |
| 12 | 0 |
Sweep one variable
Variables and how each one contributes
Probability density
What the equation gives you. Everything else feeds it.
Lower bound a
The lower bound of the estimate, in days: the least you think the quantity could be. Moves the lower end of the range; the density is zero below it. Raising it by 10 % (to 1.1) moves the result by +1.27 %, from 0.2083 to 0.211 1/day.
Most likely value m
The most likely value, in days, where the density peaks. It must lie between a and b. Moves the peak and shifts the mean towards it. Raising it by 10 % (to 3.3) moves the result by +5.26 %, from 0.2083 to 0.2193 1/day.
Upper bound b
The upper bound of the estimate, in days: the most you think it could be. Moves the upper end; a long upper tail raises the mean and the variance. Raising it by 10 % (to 9.9) moves the result by −7.77 %, from 0.2083 to 0.1922 1/day.
Test value x
The test value, in days. Moves the point at which the density or probability is read. Raising it by 10 % (to 4.4) moves the result by −8 %, from 0.2083 to 0.1917 1/day.
Worked example with the values above
- Check the side: x = 4 is above the most likely value m = 3.
- Use the falling branch: 2(b − x) = 10, divided by (b − a)(b − m) = 48.
- f = 0.20833 per day.
Assumptions and limits
- The quantity lies between a and b, with a < m < b, and the density falls linearly either side of the most likely value.
- The three values are honest estimates: the bounds are the limits you believe, not worst cases you would never see.
Common mistakes
- Choosing bounds that are too narrow: people tend to be overconfident, so the real range is usually wider than the first guess.
- Using the most likely value as the expected value: with a long upper tail the mean is above the mode.
- Adding the most likely values of several tasks to get a total: add the means instead, and add the variances for the spread.
Statistics and uncertainty
The inputs are parameters, usually fitted to data, and their estimation error carries into the result. The probability or density is a smooth function of the parameters, so the sweep chart shows how sensitive it is: where the curve is steep, a small error in a parameter makes a large difference. The formula treats the parameters as exact, so it gives no confidence interval on its own.
Where it is used
- Three-point estimates of task durations and costs in project planning and risk analysis.
- A simple input distribution for Monte Carlo analysis when only a minimum, a best guess and a maximum are known.
Maths you need
- Probability density: area under the curve gives probability
- Quadratic equations: solving for the percentile