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Topic: Random numbers
Replies: 64   Last Post: Dec 24, 2007 1:04 PM

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adder

Posts: 153
Registered: 10/3/07
Re: Random numbers
Posted: Dec 24, 2007 1:45 AM
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On Dec 23, 10:32 pm, no comment <adler.m...@gmail.com> wrote:
> On Dec 23, 9:32 pm, quasi <qu...@null.set> wrote:
>
>
>

> > On Mon, 24 Dec 2007 00:10:57 -0500, quasi <qu...@null.set> wrote:
> > >On Sun, 23 Dec 2007 19:46:21 -0800 (PST), bill <b92...@yahoo.com>
> > >wrote:

>
> > >>On Dec 22, 5:25 pm, quasi <qu...@null.set> wrote:
> > >>> On Sat, 22 Dec 2007 16:30:20 -0800 (PST), bill <b92...@yahoo.com>
> > >>> wrote:

>
> > >>> >On Dec 22, 10:16 am, quasi <qu...@null.set> wrote:
> > >>> >> On Fri, 21 Dec 2007 10:57:00 -0800 (PST), simple.pop...@gmail.com
> > >>> >> wrote:

>
> > >>> >> >On Dec 21, 11:37 pm, bill <b92...@yahoo.com> wrote:
> > >>> >> >> On Dec 21, 3:16 am, John <iamach...@gmail.com> wrote:
>
> > >>> >> >> > Given a function that returns a random number between 1-5, write one
> > >>> >> >> > that returns a random number between 1-7 for the case when it should
> > >>> >> >> > be integer and for the case it can be real.

>
> > >>> >> >> Let S be the function that generates a RN between 1 and 5. Then
>
> > >>> >> >> T = S_1 + S_2 + ... + S_7
>
> > >>> >> >> For the reals , RN_7 = T/7
>
> > >>> >> >May be this should fix it:
>
> > >>> >> >For the reals , RN_7 = 1 + (T-7)*3/14
>
> > >>> >> Yes, that fixes the range.
>
> > >>> >> But it's still biased (that is, not a unform distribution).
>
> > >>> >> quasi
>
> > >>> >The OP does not specify a uniform
> > >>> >distribution, merely the range.

>
> > >>> This has already been discussed.
>
> > >>> The obvious assumption _implicit_ in the problem, even if not unstated
> > >>> is that the resulting distribution should be uniform. Of course, it
> > >>> should have been specified, but common sense dictates that in the
> > >>> absence of the required info, to choose the natural default.

>
> > >>> If there was no preference for a distribution, there would be no need
> > >>> to use the RNG provided for the range 1 to 5. We could just always
> > >>> produce the number 3, for example. In other words, the very fact that
> > >>> an RNG for the range 1 to 5 was given as part of the problem makes it
> > >>> clear that the for the actual problem (not the OP's deficient
> > >>> statement of it), it almost certainly _was_ specified that the
> > >>> required distribution should be uniform.

>
> > >>> >RN_7 = T/7 satisfies the range 1 thru 7.
>
> > >>> So what? It's badly biased. Worse, since there is no discussion of
> > >>> bias or the lack of it, it's misleading to those unaware of the issue.

>
> > >>> >T/7 is a numner in the range 1 thru 7,
> > >>> >but is it random?

>
> > >>> Ok, but note that T/7 never exceeds 5.
>
> > >>> It's definitely not uniformly random.
>
> > >>> >If RN_7 = T mod 7 +1, the probability
> > >>> >of a correct guess is 1/7

>
> > >>> Nonsense. Do a simulation.
>
> > >Ok, for the above, I must apologize. For the _integer_ case, the
> > >calculation (T mod 7) + 1 does appear to give a uniform distribution

>
> > Hehe -- I take back part of my apology. It _is_ biased, but only
> > slightly.

>
> > The probabilities for y = (T mod 7) + 1 are as follows:
>
> > P(y=1) = .1430656
> > P(y=2) = .1430016
> > P(y=3) = .1428224
> > P(y=4) = .1426432
> > P(y=5) = .1426432
> > P(y=6) = .1428224
> > P(y=7) = .1430016

>
> > The above probabilities are exact, hence you can see a slight bias. Of
> > course, while I had originally intuited a bias, I expected it to be
> > badly biased. I was wrong about that. Only the continuous case is
> > badly biased.

>
> > quasi
>
> As to the bias, I had pointed it out quite a while ago. Any random
> variable which is a function of a fixed number n of independent
> random variables, each say uniformly distributed on the integers from
> 1 to 5, CANNOT be uniformly distributed on the integers from 1 to 7,
> for the very simple reason that 7 does not divide 5^n.


The following question may be challenging. Imagine an algorithm that
A produces a distribution which is uniform on the integers from 1 to
7, by "calling" a random number generator that produces numbers
uniformly distributed on the integers from 1 to 5. The number of
calls will also be a random variable. Let its mean be m(A). What is
the smallest possible value of m(A)? (There is nothing very special
here about 5 ann 7.)


Date Subject Author
12/21/07
Read Random numbers
Champ
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Marshall
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Phil Carmody
12/21/07
Read Re: Random numbers
Marshall
12/21/07
Read Re: Random numbers
briggs@encompasserve.org
12/21/07
Read Re: Random numbers
William Elliot
12/21/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
William Elliot
12/21/07
Read Re: Random numbers
Pubkeybreaker
12/21/07
Read Re: Random numbers
b92057@yahoo.com
12/22/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
simple.popeye@gmail.com
12/21/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
Gib Bogle
12/22/07
Read Re: Random numbers
quasi
12/21/07
Read Re: Random numbers
Marshall
12/22/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
quasi
12/22/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
quasi
12/23/07
Read Re: Random numbers
simple.popeye@gmail.com
12/23/07
Read Re: Random numbers
simple.popeye@gmail.com
12/23/07
Read Re: Random numbers
simple.popeye@gmail.com
12/23/07
Read Re: Random numbers
simple.popeye@gmail.com
12/23/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
adder
12/22/07
Read Re: Random numbers
simple.popeye@gmail.com
12/22/07
Read Re: Random numbers
Herman Rubin
12/22/07
Read Re: Random numbers
adder
12/22/07
Read Re: Random numbers
b92057@yahoo.com
12/22/07
Read Re: Random numbers
quasi
12/23/07
Read Re: Random numbers
b92057@yahoo.com
12/23/07
Read Re: Random numbers
quasi
12/23/07
Read Re: Random numbers
b92057@yahoo.com
12/24/07
Read Re: Random numbers
quasi
12/24/07
Read Re: Random numbers
quasi
12/24/07
Read Re: Random numbers
adder
12/24/07
Read Re: Random numbers
quasi
12/24/07
Read Re: Random numbers
adder
12/24/07
Read Re: Random numbers
b92057@yahoo.com

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