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Continuous vs discrete variables, return period formulae

Short answer

Continuous random variables (e.g. annual peak discharge, rainfall depth) take any value in an interval; discrete variables (e.g. number of rainy days) take distinct countable values. Return period plotting-position formulas include Weibull T=(N+1)/m, California T=N/m, and Hazen T=2N/(2m-1).

Differentiate between continuous and discrete random variables. Give examples each in hydrology. Give three formulae which are used to determine the return period.

Continuous and discrete random variables

BasisContinuous random variableDiscrete random variable
ValuesCan take any value in an interval (infinite possible values)Takes only distinct, countable values (0, 1, 2, ...)
Obtained byMeasuringCounting
Probability described byProbability density function f(x)f(x); probability is area under the curveProbability mass function P(x)P(x); probability at each value
Probability at a single valueZeroNon-zero
Cumulative functionIntegral of the PDFSum of the PMF
Examples in hydrologyAnnual peak discharge (m³/s), rainfall depth (mm), evaporation, river stageNumber of rainy days in a month, number of floods exceeding a level in 50 years, number of years with drought

Formulae for return period

With N years of record and m the rank of the event (largest = 1), the plotting position formulae are:

  1. Weibull: T=N+1mT=\dfrac{N+1}{m} (most commonly used)
  2. California: T=NmT=\dfrac{N}{m}
  3. Hazen: T=2N2m−1T=\dfrac{2N}{2m-1} (i.e. N/(m−0.5)N/(m-0.5))

Other forms are Gringorten, T=N+0.12m−0.44T=\dfrac{N+0.12}{m-0.44}, and Blom. In terms of probability, T=1/PT=1/P, where PP is the exceedance probability.

✓ Answer checked by Claude Sonnet 5.5

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