How to make 5000w 250v Free Electric Generator at Home

How to make 5000w 250v Free Electric Generator at Home

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Linear Congruential Generators

- Most simulation algorithms are based on standard random number

generators,- whose goal is to generate sequences

of numbers in

the unit interval . These are the so-called standard

pseudo-random numbers, - which can be regarded as realizations of independent and on

uniformly distributed random variables

.

- whose goal is to generate sequences
- A commonly established procedure to generate standard pseudo-random

numbers is the following linear congruential method,

As a next step we will solve the recursion equation

(1), i.e., we will show how the number that

has been recursively defined in (1) can be

expressed directly by the initial value and the

parameters , and .

- Proof
- Remarks

We will now mention some (sufficient and necessary) conditions for

the parameters , , and , respectively, ensuring

that the maximal possible period is obtained.

A proof of Theorem 3.2 using results from

number theory (one of them being Fermat’s little theorem) can be

found e.g.

- in Section 2.7 of B.D. Ripley Stochastic Simulation, J.

Wiley & Sons, New York (1987) or - in Section 3.2 of D.E. Knuth (1997) The Art of Computer

Programming, Vol. II, Addison-Wesley, Reading MA.

- We also refer to these two texts for the discussion
- of other generators for standard pseudo-random numbers like

nonlinear congruential generators, shift-register generators and

lagged Fibonacci generators as well as their combinations, - alternative conditions for the parameters , , and of

the linear congruential generator defined in (1), - ensuring the generation of sequences

whose period

is as large as possible and also exhibiting other desirable

properties.

- of other generators for standard pseudo-random numbers like
- One of those properties is
- that the points

formed by pairs

of consecutive pseudo-random numbers , are

uniformly spread over the unit square . - The following numerical examples illustrate that relatively small

changes of the parameters and can result in completely

different point patterns

.

- that the points
- Further details can be found in the text by Ripley (1987) that has

been already mentioned and in the lecture notes by H. Künsch

(ftp://stat.ethz.ch/U/Kuensch/skript-sim.ps) that also contains the

following figures.

Figure 3:

Point patterns for pairs

of consecutive

pseudo-random numbers for

Point patterns for pairs

of consecutive

pseudo-random numbers for

Figure 4:

Point patterns for pairs

of consecutive

pseudo-random numbers for

Point patterns for pairs

of consecutive

pseudo-random numbers for

Figure 5:

Point patterns for pairs

of consecutive

pseudo-random numbers for

Point patterns for pairs

of consecutive

pseudo-random numbers for

Next: Statistical Tests

Up: Generation of Pseudo-Random Numbers

Previous: Simple Applications; Monte-Carlo Estimators

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Ursa Pantle

2006-07-20