Lesson 3.8 - Probability Distributions & Return Periods

Target: ~10 min read - 25 min hands-on

Overview

scipy.stats provides ready-made distributions - Normal, Lognormal, and Gumbel (Extreme Value) among them - used constantly in engineering risk analysis. This lesson introduces return periods: a "50-year peak" doesn't mean it happens once every 50 years like clockwork - it means a 1-in-50 (2%) chance of being equaled or exceeded in any given year. We'll analyze annual peak electrical demand with the Gumbel method, the direct rehearsal for Mini-Project 3.

Why This Matters (Engineering Context)

The Gumbel Extreme Value method sizes anything driven by a rare maximum - a substation transformer against peak demand, a relief system against a worst-case surge, a structure against extreme wind, a drainage culvert against a flood. Same math, many fields.

Code-Along

from scipy import stats
import numpy as np

np.random.seed(21)

# [Electrical] synthesise a 30-year record of annual peak demand (MW)
# by drawing from a Gumbel (extreme-value) distribution
n_years = 30
annual_peak = np.round(stats.gumbel_r.rvs(loc=180, scale=25, size=n_years, random_state=21), 1)
print("Annual peak demand (MW), 30-year record:")
print(annual_peak)

# .fit() estimates the distribution parameters that best match the data
loc_fit, scale_fit = stats.gumbel_r.fit(annual_peak)
print(f"\nFitted Gumbel: loc={loc_fit:.2f}, scale={scale_fit:.2f}")

# Return period T <-> non-exceedance probability (1 - 1/T).
# .ppf (percent-point function) is the inverse CDF: probability -> value.
for T in [10, 25, 50, 100]:
    x_T = stats.gumbel_r.ppf(1 - 1 / T, loc=loc_fit, scale=scale_fit)
    print(f"  {T}-year return-period peak: {x_T:.1f} MW")

# Same idea with a Normal fit, to show the distribution choice matters
mu, sigma = stats.norm.fit(annual_peak)
print(f"\n100-yr peak, Normal fit:  {stats.norm.ppf(0.99, mu, sigma):.1f} MW")   # 0.99 = 1 - 1/100
print(f"100-yr peak, Gumbel fit:  {stats.gumbel_r.ppf(0.99, loc_fit, scale_fit):.1f} MW")
print("(Gumbel is built for extremes; Normal usually understates the tail.)")

Run it: the Gumbel return-period peaks increase with return period (10-yr < 25-yr < 50-yr < 100-yr), and the Normal fit typically gives a lower (less conservative) 100-year estimate - a concrete illustration of why distribution choice is a real engineering decision.

Practice Exercises

1. Compute the 5-year and 200-year return-period peaks using the fitted Gumbel.
2. What is the probability that the peak exceeds the 50-year value in any given year? (Hint: 1/T.)
3. Fit a Lognormal distribution (stats.lognorm.fit, with floc=0) to the same data and compare its 100-year estimate to the Gumbel and Normal results.

# Try the practice exercises here

Knowledge Check

1. What does a "50-year peak" actually mean in probability terms?
2. Which distribution family is standard for extreme-value / return-period analysis?
3. Why might a Normal fit understate the risk of a rare extreme compared to a Gumbel fit?