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RandomNumbers - AM 207 Random Numbers Spring 2013 Pavlos...

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AM 207 Random Numbers Spring 2013 Pavlos Protopapas 1 Intorduction We have introduced the concept of randomly throwing stones or randomly selecting pairs. In this lecture we will explore how computers generate random numbers. Randomness means lack of correlation. For a sequence of numbers r = r 1 , r 2 , . . . , r n we can define randomness as an asymptotic property of the series as N → ∞ . Since this is a hopeless task we can test randomness with a various tests described below. Long sequences of random numbers are needed in numerous applications, in particular methods which utilize random numbers such as Monte Carlo simulation techniques, stochas- tic optimization, cryptography calculations that simulate naturally random processes (e.g. thermal motion or radioactive decay). All these methods require fast and reliable random number sources. Many physical processes are random by nature and such processes can be used to produce random numbers. Examples are noise in semiconductor devices or throwing a dice. On the other hand computers are deterministic machines and because of that, they can not generate truly random numbers. In practice, random numbers are generated by pseudorandom number generators. These are deterministic algorithms, and consequently the generated numbers are only ”pseudo- random” and have their limitations. But for many applications, pseudorandom numbers can be successfully used to approximate real random numbers. Lets say that the probability of a random number to occur is P ( r ) and that means the probability of finding r i in the interval [ r j , r j + dr ] is P ( r ) dr . A uniform distribution means that P(r) is constant and that means all numbers are equally likely to occur. Not all random sequences are uniform. In other distributions (Normal, Poisson etc) not all numbers are equally likely to occur. 2 Random Number generators There plenty of random number generators. Most use integer arithmetic and the real numbers in (0 , 1] are produced by scaling. 2.1 Linear congruential generator The Linear congruential geneator is based on a integer recursive relation r i +1 = ( a r i + c ) mod M where a, c and M are constants. This generates a sequence r 1 , r 2 , . . . of random integers which are distributed between [0 , M - 1] (if c > 0) or between [1 , M ] (if c = 0). Each r i is i
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scaled to the interval (0 , 1) by dividing by M . The parameter M is usually equal or nearly equal to the largest integer of the computer. This determines the period P of the generator and P < M . The first number in the sequence r 1 is an input and it is called the seed .
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