quasi
Posts:
12,067
Registered:
7/15/05


Re: Random numbers
Posted:
Dec 24, 2007 2:14 AM


On Sun, 23 Dec 2007 22:32:06 0800 (PST), no comment <adler.math@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 15, write one >> >>> >> >> > that returns a random number between 17 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 + (T7)*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.
Right, I see that.
Actually, I just posted a problem relating to this issue (I had not yet read this reply).
quasi

