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This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Uniform pseudo-random variable generation

In this paragraph, our goals will be to look at, in more detail, how and whether particular types of pseudo-random variable generators work, and how, if necessary, we can implement a generator of our own choosing. Below a list of requirements is listed for our uniform random variable generator:

  • A uniform marginal distribution,
  • Independence of the uniform variables,
  • Repeatability and portability,
  • Computational speed.

Current algorithms

The generation of pseudo-random variates through algorithmic methods is a mature field in the sense that a great deal is known theoretically about different classes of algorithms, and in the sense that particular algorithms in each of those classes have been shown, upon testing, to have good statistical properties. In this section, let describe the main classes of generators, and then let make specific recommendation about which generators should be implemented.

Congruential Generators

The most widely used and best understood class of pseudo-random number generators are those based on the linear congruential method introduced by Lehmer (1951) . Such generators are based on the following formula:

U i = ( a U i 1 + c ) mod m ,

where U i , i = 1,2,... are the output random integers; U 0 is the chosen starting value for the recursion, called the seed and a , c , and m are prechosen constants.

to convert to uniform ( 0,1 ) variates, we need only divide by modulus m , that is, we use the sequence { U i / m } .

    The following properties of the algorithm are worth stating explicitly:

  • Because of the “mod m” operation (for background on modular operations, see Knuth, (1981) ), the only possible values the algorithm can produce are the integers 0,1,2,..., m 1. This follows because, by definition, x mod m is the remainder after x is divided by m .
  • Because the current random integer U i depends only on the previous random integer U i 1 once a previous value has been repeated, the entire sequence after it must be repeated. Such a repeating sequence is called a cycle , and its period is the cycle length . Clearly, the maximum period of the congruential generator is m . For given choices of a , c , and m , a generator may contain many short cycles, (see the Example 1 below), and the cycle you enter will depend on the seed you start with. Notice that the generator with many short cycles is not a good one, since the output sequence will be one of a number of short series, each of which may not be uniformly distributed or randomly dispersed on the line or the plane. Moreover, if the simulation is long enough to cause the random numbers to repeat because of the short cycle length, the outputs will not be independent.
  • If we are concern with a uniform ( 0,1 ) variates, the finest partition of the interval ( 0,1 ) that this generator can provide is [ 0,1 / m ,2 / m ,..., ( m 1 / m ) ] . This is, of course, not truly a uniform ( 0,1 ) distribution since, for any k in ( 0, m 1 ) , we have P [ k / m < U < ( k + 1 ) / m ] = 0 , not 1 / m are required by theory for continuous random variables.
  • Choices of a , c , and m , will determine not only the fineness of the partition of ( 0,1 ) and the cycle length, and therefore, the uniformity of the marginal distribution, but also the independence properties of the output sequence. Properly choosing a , c , and m is a science that incorporates both theoretical results and empirical tests. The first rule is to select the modulus m to be “as large as possible”, so that there is some hope to address point 3 above and to generate uniform variates with an approximately uniform marginal distribution. However, simply having m large is not enough; one may still find that the generator has many short cycles, or that the sequence is not approximately independent. See example 1 below.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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