Random-Number Generation

[Pages:46]Random-Number Generation

Raj Jain Washington University Saint Louis, MO 63130

Jain@cse.wustl.edu

Audio/Video recordings of this lecture are available at:



Washington University in St. Louis

CSE574s

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?2008 Raj Jain

Overview

! Desired properties of a good generator ! Linear-congruential generators ! Tausworthe generators ! Survey of random number generators ! Seed selection ! Myths about random number generation

Washington University in St. Louis

CSE574s

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?2008 Raj Jain

Random-Number Generation

! Random Number = Uniform (0, 1) ! Random Variate = Other distributions

= Function(Random number)

Washington University in St. Louis

CSE574s

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?2008 Raj Jain

A Sample Generator

! For example,

! Starting with x0=5:

! The first 32 numbers obtained by the above procedure 10, 3, 0, 1, 6, 15, 12, 13, 2, 11, 8, 9, 14, 7, 4, 5 10, 3, 0, 1, 6, 15, 12, 13, 2, 11, 8, 9, 14, 7, 4, 5.

! By dividing x's by 16: 0.6250, 0.1875, 0.0000, 0.0625, 0.3750, 0.9375, 0.7500, 0.8125, 0.1250, 0.6875, 0.5000, 0.5625, 0.8750, 0.4375, 0.2500, 0.3125, 0.6250, 0.1875, 0.0000, 0.0625, 0.3750, 0.9375, 0.7500, 0.8125, 0.1250, 0.6875, 0.5000, 0.5625, 0.8750, 0.4375, 0.2500, 0.3125.

Washington University in St. Louis

CSE574s

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?2008 Raj Jain

Terminology

! Seed = x0 ! Pseudo-Random: Deterministic yet would pass randomness

tests ! Fully Random: Not repeatable ! Cycle length, Tail, Period

Washington University in St. Louis

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?2008 Raj Jain

Desired Properties of a Good Generator

! It should be efficiently computable. ! The period should be large. ! The successive values should be independent and

uniformly distributed

Washington University in St. Louis

CSE574s

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?2008 Raj Jain

Types of Random-number Generators

! Linear congruential generators ! Tausworthe generators ! Extended Fibonacci generators ! Combined generators

Washington University in St. Louis

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?2008 Raj Jain

Linear-Congruential Generators

! Discovered by D. H. Lehmer in 1951 ! The residues of successive powers of a number have good

randomness properties.

Equivalently,

a = multiplier m = modulus

Washington University in St. Louis

CSE574s

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?2008 Raj Jain

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