strange problems with Random

*To*: mathgroup at smc.vnet.net*Subject*: [mg47812] strange problems with Random*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 27 Apr 2004 04:48:05 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Recently Mark Coleman wrote a message with the subject "Differences in Random Numbers" in which he described his experience of getting statistically incorrect answers when using Random[UniformDistribution[0,1]] instead of simple Random[]. Jens Kuska and Bill Rowe pointed out that the package ContinuousDistributions contains the code UniformDistribution/: Random[UniformDistribution[min_:0, max_:1]] := Random[Real, {min, max}] and that seemed to be it. However, I have now encountered a problem that makes me feel strongly that there is more to this problem than it seemed to me at first. I have been doing some Monte Carlo computations of values of of financial options for the Internet based course on this subject I am teaching for Warsaw University. In the case of the simplest options, the so called vanilla european options, an exact formula (due to black and Scholes) is known. Neverheless, computations with Mathematica using the function Random[NormalDistribution[mu,sigma]] (for some fixed mu and sigma) have been giving answers which are consistently higher than the Black-Scholes value. On the other hand, when Random[NormalDistribution[mu,sigma]] is replaced by the following code using a well known algorithm due to Marsaglia: RandomNormal=Compile[{mu, sigma}, Module[{va = 1., vb, rad = 2.0, den = 1.}, While[rad ³ 1.00, (va = 2.0*Random[] - 1.0; vb = 2.0*Random[] - 1.0; rad = va*va + vb*vb)]; den = Sqrt[-2.0*Log[rad]/rad]; mu + sigma*va*den]]; the answers agree very closely with the Black-Scholes ones. I am not sure if this problem is related to the one mark mentioned but it seems quite likely. One could in principle test it by replacing Random[] by Random[Uniformdistribution[0,1]] (after loading the Statistics`ContinuousDistributions`) package, but if one does that the code will no longer compile. That makes the function to slow for tests that are sufficiently precise, so I have not tested it. In any case, I am sure that using Random[Normal ] consistently gives wrong answers. I looked at the Statistics`NormalDistributionPackage` but everything looks O.K. to me. I now tend to believe that the problem lies with Random itself, that is that Random called with any parameters or perhaps some particular parameters may not be behaving correctly. Random[] called with no parameters does not seem to suffer from any such problems. Andrzej Kozlowski Chiba, Japan http://www.mimuw.edu.pl/~akoz/