Probability distributions in PRQ

Every risk in the PRQ register has an impact distribution: a family of values the impact could take, with their relative likelihoods. PRQ offers six families. Three are the workshop defaults — triangular, PERT, and fixed — chosen because they cover the practical spread of project-risk impact shapes without requiring elicitation any more elaborate than a low / most likely / high triple. Three more sit on the specialist shelf — lognormal, normal, and uniform — for the cases where the shape itself is the point.

The job of this article is to make the choice between them deliberate. The wrong distribution exaggerates or hides tail risk; the right one matches the way the underlying impact actually behaves.

Triangular — the workhorse

Specified by three numbers: min, mode, and max. Every value in [min, max] is possible; the probability density is a triangle peaking at the mode and falling linearly to zero at the bounds. Mean is (min + mode + max) / 3; the distribution is symmetric only when mode = (min + max) / 2.

Reach for triangular when:

• You can elicit a low / most-likely / high triple from a workshop, but you don't have data on the inner shape.
• You want the bounds to be hard limits — values strictly outside [min, max] are impossible by construction.
• The default. If you can't articulate a positive reason to use something else, triangular is the safe choice.

Watch out for: triangular puts more weight near the bounds than most real-world impacts warrant, because the density falls away in a straight line rather than tapering. How much more depends on where the mode sits. For a symmetric triangular, the chance of landing in the top tenth of the range is 2.0%; on the sample register's own vendor-outage distribution (50 / 120 / 350) it is 1.3%; and it only reaches 4% when the mode sits three quarters of the way up. Modest numbers, but a genuinely rare worst case is rarer than any of them.

PERT — when extreme values are unlikely but not impossible

Same three parameters as triangular (min, mode, max) but the density is a smooth bell-shaped curve concentrated near the mode. Originally from the US Navy's Program Evaluation and Review Technique (1958), which is where the name comes from. Mean is (min + 4·mode + max) / 6 — note the 4× weight on the mode, which captures the assumption that most outcomes cluster near the mode.

Reach for PERT when:

• You believe the mode genuinely is the most likely outcome by a significant margin and the bounds are rare-tail, not plausible-tail.
• The impact is the result of summing many small effects (central- limit-style behaviour). Schedule-driven cost overruns often fit this — many small slip-causes, summing to a near-mode total most of the time.
• You want the standard "expected value" interpretation: PERT's mean weights the mode 4× the bounds, which most stakeholders intuitively expect.

Watch out for:PERT understates real tail risk compared to triangular. If you genuinely think the worst case could happen with appreciable probability (say, >5%), triangular is more honest.

Fixed — when the impact isn't actually a distribution

A single value. Specified as value. Used when the impact, given that the risk triggers, is essentially known — contractually fixed penalties, predictable per-incident costs, regulatory fines with a published schedule.

Reach for fixed when:

• The impact is a price tag, not a range. "$50k regulatory fine if breach is reported" is fixed; "customer notification costs" is not.
• You want to keep the model honest about which risks have uncertain impact and which don't. A register full of fixed impacts produces a degenerate Monte Carlo run; a mix tells a richer story.

Watch out for: tempting to use fixed for "everything we don't have data on" because it's the simplest input. That collapses the variance the simulation exists to capture.

Eliciting parameters in a workshop

The practical bottleneck on PRQ accuracy isn't the engine — it's the per-risk distributions. A small playbook that works:

1. Ask for the most likely value first. "If this risk triggers, what's the single number you'd put as the central guess?" That's the mode. Don't anchor with a range yet.
2. Ask for the optimistic value second. "What's the value below which there's only a 5–10% chance the impact falls?" That's the min. (Not the absolute minimum — anchoring on the absolute extreme produces narrow distributions.)
3. Ask for the pessimistic value last. "What's the value above which there's only a 5–10% chance the impact falls?" That's the max.
4. Sanity-check the spread. A max-to-mode ratio above ~5× usually means either the SME has factored in low- probability tail events into the bounds (split the risk), or they're under-confident (push back).
5. Pick the family last. Triangular by default; PERT if the SME reacts strongly to "could the worst case actually happen?" with "no, that's basically impossible".

The specialist shelf: lognormal, normal, uniform

Reach past the three-point families only when the shape is the point. The discipline argument for triangular and PERT still stands — restricting the input vocabulary keeps the conversation about the numbers — so treat these as tools with specific jobs rather than upgrades.

• Lognormal is the conventional shape for cost impacts with a genuinely heavy upside tail: multiplicative effects (delay begets delay begets penalty) produce it naturally, and unlike a three-point estimate it has no hard maximum, so the tail keeps going. The classic objection was elicitation — nobody has an opinion about "sigma of the log" — so PRQ asks for the median (the middle outcome) and the P90 (a bad but plausible outcome), both in dollars, which pin the distribution exactly. If your honest P90 sits far above your honest maximum-credible value, a bounded three-point family was the truer statement.
• Normal is symmetric around its mean and unbounded in both directions, which means it can produce negative impacts, and the model will happily sum them. That is occasionally right (an outcome that can genuinely come in under as often as over), and mostly it isn't for cost risk. Prefer lognormal unless the deviations really are symmetric and small relative to the mean.
• Uniform says every value between min and max is equally likely, which is a stronger claim than it sounds — it asserts that knowing nothing inside the bounds is the honest position. Genuine ignorance-within-hard-bounds exists (a contractual penalty that will land somewhere in a fixed range), but usually a workshop knows more than that, and a three-point estimate should say so.

Beta and arbitrary user-specified distributions stay unexposed. If you need a shape PRQ doesn't model, the right move is usually to split the risk into two with different bounds, or to elicit the impact as the sum of several smaller risks with their own distributions.

How PRQ samples them

Triangular samples by closed-form inverse transform (draw a uniform, look up the quantile). PERT is parameterised as a four-parameter beta and sampled via gamma deviates (Marsaglia–Tsang). Normal uses Box–Muller; lognormal exponentiates a normal; uniform and fixed are what they say. Where a single uniform must map to a single value — the seeded paired runs in Event Risk — the engine also carries a true beta quantile function, so PERT inverts exactly rather than being approximated. All of it runs in a Web Worker so the page stays responsive.