Library / Probability distributions / PRD-38
Probability distributions
Gumbel maxima: probability of staying below a threshold
Probability that the largest value in a block, such as the strongest gust of a year, stays at or below x under a Gumbel model with location m and scale s.
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- PRD-38
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- v1
- Field
- Probability distributions
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- F
The Gumbel distribution function e^(−e^(−z)) with z = (x − m)/s gives the probability that a block maximum, such as the strongest gust of a year, stays at or below x. Its complement is the exceedance probability; its reciprocal is the return period in blocks.
Probability that the largest value in a block, such as the strongest gust of a year, stays at or below x under a Gumbel model with location m and scale s.
Calculator
- Exceedance probability, 1 − F0.3078
- Return period in blocks, 1/(1 − F)3.249 blocks
- Level with a 50-block return period, m − s ln(−ln(1 − 1/50))31.71 m/s
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.
- The chance that the block maximum stays at or below 23 m/s is 69.2 %, a return period of about 3.25 blocks.
- The distribution is right-skewed: exceeding a high level becomes rarer more slowly than for a normal model, so design values need care.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Threshold x | 5 to 50 m/s | 3.507e-65 | 1 | 1 | +10% | +21.8 % |
| m Location m | 0 to 40 m/s | 2.876e-126 | 0.9995 | 0.9995 | +10% | −29.4 % |
| s Scale s | 0.5 to 10 m/s | 0.4767 | 0.9975 | 0.5208 | +10% | −3.44 % |
Sweep of Threshold x
| Threshold x (m/s) | Probability maximum ≤ x |
|---|---|
| 5 | 3.507e-65 |
| 10.63 | 1.305e-10 |
| 16.25 | 0.03049 |
| 21.88 | 0.5855 |
| 27.5 | 0.9212 |
| 33.13 | 0.9875 |
| 38.75 | 0.9981 |
| 44.38 | 0.9997 |
| 50 | 1 |
Sweep one variable
Variables and how each one contributes
Probability maximum ≤ x
What the equation gives you. Everything else feeds it.
Location m
The location, in m/s: the mode, the most likely block maximum. Shifts the whole distribution. Raising it by 10 % (to 22) moves the result by −29.4 %, from 0.6922 to 0.4884.
Scale s
The scale, in m/s: it sets the spread of the block maxima. Widens the distribution and lengthens the upper tail. Raising it by 10 % (to 3.3) moves the result by −3.44 %, from 0.6922 to 0.6684.
Threshold x
The threshold, in m/s, at which the result is read. Moves the point at which the density or probability is read. Raising it by 10 % (to 25.3) moves the result by +21.8 %, from 0.6922 to 0.8429.
Worked example with the values above
- Standardised value: z = (23 − 20)/3 = 1.
- Inner exponential: e^(−z) = 0.36788.
- F = e^(−0.36788) = 0.6922.
Assumptions and limits
- Each value is the maximum of many roughly independent observations in a block of equal length, such as a year.
- The location and scale come from a fit to block maxima of the same kind.
Common mistakes
- Mixing blocks of different lengths in one fit.
- Treating the probability as a return period: a non-exceedance probability F corresponds to a return period of 1/(1 − F) blocks.
- Extrapolating far beyond the data, where the choice of tail model dominates.
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
- Design wind speeds, flood levels and wave heights from annual maxima.
- Maximum loads and extreme temperatures in structural and thermal design.
Maths you need
- Exponential function: e raised to a power
- Return period: average time between exceedances