---
title: "Continuous vs discrete variables, return period formulae"
subject: "Engineering Hydrology"
chapter: 6
key: "semester_5/engineering_hydrology:continuous-discrete-variables-return-period"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/continuous-discrete-variables-return-period
site: BCT Engineering
---

# Continuous vs discrete variables, return period formulae

Engineering Hydrology · Chapter 6: Flood Hydrology

- Asked in: 2073 Shrawan, 1+2+3 marks
- Video: [Continuous vs Discrete Data](https://www.youtube.com/watch?v=cz4nPSA9rlc) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/continuous-discrete-variables-return-period
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6

## Question

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

## 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).

## Answer

#### Continuous and discrete random variables
| Basis | Continuous random variable | Discrete random variable |
|---|---|---|
| Values | Can take any value in an interval (infinite possible values) | Takes only distinct, countable values (0, 1, 2, ...) |
| Obtained by | Measuring | Counting |
| Probability described by | Probability density function $f(x)$; probability is area under the curve | Probability mass function $P(x)$; probability at each value |
| Probability at a single value | Zero | Non-zero |
| Cumulative function | Integral of the PDF | Sum of the PMF |
| Examples in hydrology | Annual peak discharge (m³/s), rainfall depth (mm), evaporation, river stage | Number 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:** $T=\dfrac{N+1}{m}$ (most commonly used)
2. **California:** $T=\dfrac{N}{m}$
3. **Hazen:** $T=\dfrac{2N}{2m-1}$ (i.e. $N/(m-0.5)$)

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A continuous random variable differs from a discrete one in that:
   A. It can take any value in an interval and is measured
   B. It takes only countable values
   C. It has a non-zero probability at a single value
   D. It is described by a mass function
   Answer: A. Continuous variables use a PDF and have zero probability at one value. (Key point: Continuous vs discrete)

2. A discrete variable example is:
   A. Annual peak discharge
   B. Number of rainy days in a month
   C. Rainfall depth
   D. River stage
   Answer: B. Counting gives discrete values. (Key point: Hydrology example)

3. The Weibull formula for return period is:
   A. T = N/m
   B. T = 2N/(2m-1)
   C. T = (N+1)/m
   D. T = m/N
   Answer: C. California is N/m; Hazen is 2N/(2m-1). (Key point: Weibull)

## Related in Engineering Hydrology

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