Turn a uniform number to normal random numbers

[Yves]

Hi,

I read from Mark Joshi's Concept of Mathematical Finance pg 178 "..there is a simple method which gives reasonable, but not great, approximation is to simply add together 12 uniform variables and subtract 6. The results has correct mean, variance and third moment."

Could someone explain this idea? How can I find out about quick method?

Thanks.

[Gordon Sande]

The uniform on 0-1 has a variance of 1/12 and a mean of 1/2.
So the sum of 12 of them has a variance of 1 and a mean of 6.
Subtract 6 for a variance of 1 and a mean of 0 as wanted for
a standard normal. The result is symmetric so the third moment
(and all the odd moments as well!) will be zero.

A quick and dirty plausible approximation as promised. Notice that
there will be no values below -6 or above +6 at all. The approximation
is too short tailed for many other purposes. Remember quick and dirty
and not great were the words in the description.

[Luis A. Afonso]

The EXACT method to get a pair of INDEPENDENT Normal (mean, sigma) Values (X1, X2) from Uniform pair (RND1, RND2) is:

U = SQR ( -2 * LOG (RND1 ) )

X1 = mean + sigma * U * COS (2 * pi * RND2)
X2 = mean + sigma * U * SIN (2 * pi * RND2)

________pi = 4 * ATN (1)

This is the Box-Muller algorithm.

______
Luis A. Afonso

[John Smith]

Luis,

You never bothered to finish discussing your last error before you committed another one. Please pay attention.

On Nov 11, 4:22 am

You wrote:

"From this values we obtain the CONFIDENCE INTERVALS of the two-tailed tests relative to the probabilities 99%, 98%, 95% the parameter be inside."

Either the parameter is INSIDE or OUTSIDE. The probability that the parameter is inside is either 100% or 0%. Same for the probability that the parameter is outside.

Please defend your assertion that there can be a 99% probability that a parameter is inside the interval.

John

[Luis A. Afonso]

RES NON VERBA

The QUADRUPLE´S DUO dared to fight the evidence:

1) The Empirical Distribution Function is fully able to provide the Critical Values of a Sample Statistics (as long as the random samples are attainable),

2) This method is rigorously Model Exact. (On contrary the usual ones are only mere approximations, in general).

3) The difference between the Empirical and the Exact Distribution Functions (Cumulative Density Function) can be made as short as we wish do assert the Dvoretzky- Kiefer-Wolfowitz inequality,

4) Quoting my post:
*** In fact the Dvoretzky-Kiefer-Wolfowitz assures us that ________p(d>e)<= 2*EXP(-2*n*e*e). Suppose that n=1E7 , e= 0.0005. Then: 2*EXP (-5) = 0.013 ***

5) These are FACTS: OPINIONS are unable to put down them. Not theoretically based arguments worth ZERO (RES NON VERBA).

6) This procedure was used in the paper (everybody can read):
_________Un nouveau test pour la distribution uniforme (L. Amaral Afonso, P. Duarte) was published (after a rigorous and accurate appreciation by the REFEREE TEAM) in the recognized Journal RSA (Revue de Statistique Appliquee), tome 40, nº1, (1992).

7) The TEAM should write a paper submitted to RSA (or JASA, if unable to write French) proving that I was wrong, and thousands of Statisticians too that used, since 1967*, the method for this purpose. While expecting this action WE, THE READERS, ARE FREE TO CLASSIFY THEM AS NOBODY IN STATISTICS,

* Hurbert W. Lilliefors:
ON THE KOLMOGOROV-SMIRNOV TEST FOR NORMALITY WITH MEAN AND VARIANCE UNKOWNS.
Journal of the American Statistical Association, volume 62, number 318, JUNE 1968.
_____
Luis Amaral Afonso

[Herman Rubin]

In article <13104907.1195060592...@nitrogen.mathforum.org>,
Yves <sunde...@yahoo.com> wrote:
>Hi,

>Thanks.

The idea is that the distribution is close to standard normal;
it has the right mwan and variance, and the difference of the
densities is "small".

It is not quick; there are quicker and more accurate methods.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

[John Smith]

Luis,

No professional statistician believes that a parameter has 99% chance of falling in an interval. How about if I call up Lilliefors and ask him?

[Yves]

Thanks all, for your replies.

What are some of the quicker and accurate methods?

[Ray Koopman]

On Nov 14, 6:09 pm, Yves <sunder_1...@yahoo.com> wrote:
> Thanks all, for your replies.
>
> What are some of the quicker and accurate methods?

See Luc Devroye, Non-Uniform Random Variate Generation.
http://cg.scs.carleton.ca/~luc/rnbookindex.html

[David Winsemius]

Yves <sunde...@yahoo.com> wrote in
news:18472987.1195092592...@nitrogen.mathforum.org:

> Thanks all, for your replies.
>
> What are some of the quicker and accurate methods?
>

Not sure about quicker, but more economical in the random number generation
steps, anyway. If you have two uniform "random" numbers in (0,1), U1 and
U2, you can get two "random" Normals, X1 abd X2, with this transformation:

X1=SQRT(-2*LN(U1))*COS(2*pi*U2)
X2=SQRT(-2*LN(U1))*SIN(2*pi*U2)

source: Johnson, Kotz, Balakrishnan. "Continuous Univariate Distributions,
vol 1"

The machine time will depend on the efficiency of the log and trig
functions.

--
David Winsemius

[Herman Rubin]

In article <fhfpu0$2o...@odds.stat.purdue.edu>,

Herman Rubin <hru...@odds.stat.purdue.edu> wrote:
>In article <13104907.1195060592...@nitrogen.mathforum.org>,
>Yves <sunde...@yahoo.com> wrote:
>>Hi,

>>I read from Mark Joshi's Concept of Mathematical Finance pg 178 "..there is a simple method which gives reasonable, but not great, approximation is to simply add together 12 uniform variables and subtract 6. The results has correct mean, variance and third moment."

>>Could someone explain this idea? How can I find out about quick method?

>>Thanks.

>The idea is that the distribution is close to standard normal;
>it has the right mwan and variance, and the difference of the
>densities is "small".

>It is not quick; there are quicker and more accurate methods.

Rubin, Herman and Johnson, Brad (2006)
Efficient generation of exponential and normal deviates
Journal of Statistical Computation and Simulation, 76, 509-518
CISid: 261782

This gives one method of doing it, and also refers to
the Marsaglia inverse ziggurat method, which is another.

[Luis A. Afonso]

*** Date: Nov 19, 2007 11:28 AM
Author: David Winsemius
Subject: Re: Turn a uniform number to normal random numbers

source: Johnson, Kotz, Balakrishnan. "Continuous Univariate Distributions, vol 1"The machine time will depend on the efficiency of the log and trig functions. –
David Winsemius ****

MY RESPONSE

David
If you took attention to my
Nov 14, 2007 2:05 PM post
You should admit that I answered the OP in a more general form you posted.
Echoes are useless: isn’t it?

*********

Luis Amaral Afonso

[David Winsemius]

"Luis A. Afonso" <lic...@hotmail.com> wrote in
news:16233415.1195502727...@nitrogen.mathforum.org:

> *** Date: Nov 19, 2007 11:28 AM
> Author: David Winsemius
> Subject: Re: Turn a uniform number to normal random numbers

> MY RESPONSE
>
> David
> If you took attention to my
> Nov 14, 2007 2:05 PM post

Until now I had not. You and the rest of the audience are free to speculate
on why that might be so.

> You should admit that I answered the OP in a more general form you
> posted.
> Echoes are useless: isn’t it?

I freely admit such, You should be the expert in useless posting, that is
for sure.

--
David

[Luis A. Afonso]

David

FACTS ARE FACTS

My post dated Nov 14, 2007 2:05 PM (presented in this thread) a solution that is more general than yours Nov 19, 2007 11.28 AM one.
I DIDN´T say you copied me: only that your post is useless.
Furthermore I identified the algorithm, Box- Muller’s, an useful information to the OP for ulterior self checking.

Be aware: those that prefer OPPINIONS to FACTS are incapable to perform whatsoever in SCIENCE.

****
Luis Amaral Afonso