---
title: "Properties of random numbers"
subject: "Simulation and Modeling"
chapter: 6
key: "semester_8/simulation_and_modeling:rn-properties"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-properties
site: BCT Engineering
---

# Properties of random numbers

Simulation and Modeling · Chapter 6: Random Number

- Asked in: 2074 Bhadra, 3 marks
- Video: [Properties Of Random Numbers](https://www.youtube.com/watch?v=-XMT9FDs9JY) — Vasanthamma H
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-properties
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6

## Question

What are the properties of random numbers?

## Short answer

Random numbers must be uniform on [0,1) (density f(R)=1, mean 1/2, variance 1/12) and independent (no correlation). Good generators also have maximum density, long period, reproducibility with a seed, efficiency and portability. Uniformity is tested by chi‑square or K‑S, independence by runs, autocorrelation, gap and poker tests.

## Answer

A sequence of random numbers R1, R2, R3, ... used in simulation must have two essential statistical properties, **uniformity** and **independence**, plus some practical properties of a good generator.

1. **Uniformity.** Each number is equally likely to fall in any sub-interval of [0, 1). If the interval is divided into n equal classes, each class should receive about N/n of the N numbers. The density is f(R) = 1 for 0 ≤ R ≤ 1, so E(R) = 1/2 and Var(R) = 1/12.
2. **Independence.** The value of one number must not depend on any earlier or later number. There should be no pattern, trend or correlation in the sequence, so knowing R1 tells nothing about R2.
3. **Maximum density.** Values should cover [0, 1) closely, so the generator should have a very large number of distinct values (small gaps between possible values).
4. **Long (maximum) period.** The sequence should not repeat for a very long time. Ideally the period is as large as the modulus m (full period).
5. **Reproducibility.** The same seed must give the same sequence. This is needed for debugging and for comparing alternative designs with the same random inputs.
6. **Efficiency and portability.** The generator should be fast, need little memory, and give the same results on different computers.

Uniformity is checked with the chi-square or Kolmogorov-Smirnov test. Independence is checked with the runs test, autocorrelation test, gap test and poker test.

## Check yourself

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

1. Which are the two essential statistical properties of random numbers?
   A. Long period and reproducibility
   B. Uniformity and independence
   C. Speed and portability
   D. Maximum density and efficiency
   Answer: B. Uniformity and independence are the essential statistical properties; the others are practical properties of a generator. (Key point: Essential properties)

2. What does independence of random numbers mean?
   A. Every number equals the previous one
   B. The numbers lie only in half of [0,1)
   C. No number depends on earlier or later numbers; no pattern or correlation
   D. The numbers must be integers
   Answer: C. Knowing R1 should tell nothing about R2; there must be no pattern, trend or correlation. (Key point: Independence)

3. Why is reproducibility (same seed gives same sequence) desirable?
   A. It makes the numbers truly random
   B. It allows debugging and comparison of alternative designs with the same inputs
   C. It increases the period
   D. It removes the need for tests
   Answer: B. Replaying the same stream is needed for debugging and fair comparison of designs. (Key point: Reproducibility)

4. Which test checks uniformity?
   A. Gap test
   B. Poker test
   C. Chi-square or K-S test
   D. Runs test
   Answer: C. Uniformity is checked with chi-square or K-S; independence with runs, autocorrelation, gap and poker tests. (Key point: Testing)

## Related in Simulation and Modeling

- [Chi-square test on 100 random numbers](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Achi-square-uniformity-100-numbers) — Simulation and Modeling, chapter 6
- [Random number properties and Gap test](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-gap-test) — Simulation and Modeling, chapter 6, asked 4×
- [Random number properties; KS test algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-ks-algorithm) — Simulation and Modeling, chapter 6
- [Define random variables; properties](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-define-properties-variables) — Simulation and Modeling, chapter 6
- [Random number; problems in generating pseudo-random](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-problems-pseudo-random) — Simulation and Modeling, chapter 6
- [Pseudo-random; Kolmogorov-Smirnov numerical](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aks-test-uniform-numerical) — Simulation and Modeling, chapter 6

## Related in other subjects

- [Continuous random variable; pdf properties](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acontinuous-rv-pdf-properties) — Probability and Statistics, chapter 3