Library / Probability distributions / PRD-40
Probability distributions
Gumbel maxima: expected maximum
Mean of a Gumbel block maximum: the location plus the Euler–Mascheroni constant γ times the scale.
- Ref
- PRD-40
- Rev
- v1
- Field
- Probability distributions
- Result
- E[X] [m/s]
The mean block maximum is the location plus γs, where γ ≈ 0.5772 is the Euler–Mascheroni constant. The mean exceeds the location (the mode), because of the long upper tail.
Mean of a Gumbel block maximum: the location plus the Euler–Mascheroni constant γ times the scale. The mean lies above the location.
Calculator
- Mode (the location)20 m/s
- Standard deviation, πs/√63.848 m/s
- Median, m − s ln(ln 2)21.1 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 average yearly maximum is 21.73 m/s, 1.73 m/s above the most likely maximum of 20 m/s.
- The mean rises one-for-one with the location and by 0.58 for each unit of scale.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Location m | 0 to 40 m/s | 1.732 | 41.73 | 40 | +10% | +9.2 % |
| s Scale s | 0.5 to 10 m/s | 20.29 | 25.77 | 5.484 | +10% | +0.797 % |
Sweep of Location m
| Location m (m/s) | Expected block maximum (m/s) |
|---|---|
| 0 | 1.732 |
| 5 | 6.732 |
| 10 | 11.73 |
| 15 | 16.73 |
| 20 | 21.73 |
| 25 | 26.73 |
| 30 | 31.73 |
| 35 | 36.73 |
| 40 | 41.73 |
Sweep one variable
Variables and how each one contributes
Expected block maximum
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 +9.2 %, from 21.73 to 23.73 m/s.
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 +0.797 %, from 21.73 to 21.9 m/s.
Worked example with the values above
- Euler–Mascheroni constant: γ = 0.57722.
- Multiply by the scale: γs = 1.7316 m/s.
- E[X] = 20 + 1.7316 = 21.732 m/s.
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
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
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