Library / Probability distributions / PRD-06
Probability distributions
Triangular estimate: cumulative probability
Probability that a triangular quantity is at or below x.
- Ref
- PRD-06
- Rev
- v1
- Field
- Probability distributions
- Result
- F
F is the area under the triangular density up to x. Below the most likely value it grows as the square of the distance from a; above it, it approaches 1 as the square of the distance from b. The two branches meet at m, where F = (m − a)/(b − a). Zero below a and one above b.
Probability that a triangular quantity is at or below x. It is zero below a, one above b, and the two branches meet at the most likely value m. The first form applies up to m and the second above it.
Calculator
- F at the most likely value m, (m − a)/(b − a)0.25
- Probability above x, 1 − F0.5208
- 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.
- There is a 47.9 % chance that the value is at or below 4 days, and 52.1 % that it is above.
- Because the upper tail is longer than the lower one, the chance of finishing by the most likely value is only 25 %, well below one half.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Test value x | 0 to 12 days | 0 | 1 | 1 | +10% | +16.7 % |
| b Upper bound b | 3.5 to 20 days | 0.2074 | 1 | 0.7926 | +10% | −9.6 % |
| m Most likely value m | 1.5 to 8.5 days | 0.15 | 0.5833 | 0.4333 | +10% | −5.72 % |
| a Lower bound a | −5 to 2.5 days | 0.359 | 0.7024 | 0.3434 | +0.15 days | −2.08 % |
Sweep of Test value x
| Test value x (days) | Probability at or below x |
|---|---|
| 0 | 0 |
| 1.5 | 0.01563 |
| 3 | 0.25 |
| 4.5 | 0.5781 |
| 6 | 0.8125 |
| 7.5 | 0.9531 |
| 9 | 1 |
| 10.5 | 1 |
| 12 | 1 |
Sweep one variable
Variables and how each one contributes
Probability at or below x
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 −1.38 %, from 0.4792 to 0.4726.
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 −5.72 %, from 0.4792 to 0.4518.
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 −9.6 %, from 0.4792 to 0.4332.
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 +16.7 %, from 0.4792 to 0.5592.
Worked example with the values above
- Check the side: x = 4 is above m = 3.
- Falling branch: 1 − (b − x)²/((b − a)(b − m)) with (b − x)² = 25.
- F = 0.47917.
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