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Re: generating random number

  • To: mathgroup at smc.vnet.net
  • Subject: [mg34470] Re: [mg34448] generating random number
  • From: "Johannes Ludsteck" <johannes.ludsteck at wiwi.uni-regensburg.de>
  • Date: Thu, 23 May 2002 03:32:09 -0400 (EDT)
  • Organization: Universitaet Regensburg
  • Sender: owner-wri-mathgroup at wolfram.com

Dear Ester,
if your random variable ist continuous,
the simplest way to obtain a random number distributed
according to an arbitrary PDF is to use the inversion
method: If F[x] is the distribution function of the random
variable x, an G[z] is the inverse of F
(the solution of z = F[x]),
the you obtain your pseudo random realization of x by
application of the inverse to a uniform random variable
G[Random[]]
in the same way you get an n-array of realizations with
Map[G,Table[Uniform[],{n}].

Here is the simple proof for the validity of the transformation:
If u ~ Uniform[0,1], then the
distribution function of u is H[u] = u
x ~ F[x];
v:= G[u];
P[v < z] = P[G[u] < z] = P[u < F[z]] = H[F[z]] = F[z]

The only one problem with this approach: you must
know the distribution function. If it is not possible to
integrate the density (PDF), you have to integrate it
numerically. If this is to slow, try generating a
table of the CDF, interpolate it and use the interpolation
instead of the distribution.

If this does not work or if your random variable is
discrete or mixed, have a look at the book
Luc Devroye: Non-Uniform Random Variable Generation.

Or use the package MathStatica (contained in the book)
Rose & Smith: Mathematical Statistics with Mathematica.

Best regards,
	Johannes Ludsteck

On 22 May 2002, at 2:46, ester wrote:

> 
>  dear Sirs,
>  my name is Ester Piedipalumbo and I am a post doc at the phisics dep. of
>  Naples University.
>  I would like to know if there is something for generating random number
>  according an arbitary PDF (where arbitrary means that it is not standard,
>  but I know, of cource)
>  Thank a lot
>  best regrds
>  Ester
> 
> 
> 
> 



<><><><><><><><><><><><><><><><><><>
Johannes Ludsteck
Institut fuer Volkswirtschaftslehre
Lehrstuhl Prof. Dr. Moeller
Universitaet Regensburg
Universitaetsstrasse 31
93053 Regensburg
Tel +49/0941/943-2741


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