Re: Generating Gaussian distributed random variable using C++

On Aug 2, 10:13=A0am, Rune Allnor <all...@xxxxxxxxxxxx> asked:


What's the problem? That the sum of random variables tends
towards a Gaussian dirstibution?

Yes. The central limit theorem requires certain conditions
on the random variables. It certainly is not true in all cases
that

True. I should have been more specific.

"The central limit theorem ... <gratuitous comment deleted>....
says the sum of a reasonably large bunch of random numbers looks
awfully Gaussian"

as steveu stated. This point has been discussed previously
in this newsgroup, and several people have noted, for example,
that the sum of n Cauchy random variables is a (scaled) Cauchy
random variable, and doesn't "look Gaussian".


Or the specific numbers?

To a certain extent, yes. I don't know what misconception
steveu was referring to when he wrote

"12 is a popular number to sum due to a misconception"

but 12 is often used because a random variable uniformly
distributed on (0, 1) has variance 1/12. Thus the sum of
12 such independent uniform variates has variance 1.
Subtracting off the mean (6), gives a zero-mean, unit
variance random variable whose density is a good
approximation to a unit Gaussian density. (One could,
of course, sum together a different number of uniform
variates, but then more complicated scaling is required.)
**However**, the random variable thus generated (from
the sum of 12 independent random variables) is
limited to have values in the range (-6, 6) and is not
necessarily of much help when (for example) **very
small** error probabilities are to be determined via
simulation: the error events in question might never
occur during the simulation.

Various people will tell you 12 has a magic quality, which is doesn't. It
just makes the scaling trivial. It is sometimes necessary to use far more
than 12 numbers to get a satisfactory result, but it still works.

I'm not sure what you are referring to about small error probabilities in
simulations? Any simulation of very small error probabilities is fraught
with issues. The required length of the simulation to see a rare event a
sufficient number of times being the most painful. I can't think of a
specific issue related to this method of generating a Gaussian
distribution, though.

Steve

.

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