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


Re: Random numbers
Posted:
Dec 24, 2007 12:10 AM


On Sun, 23 Dec 2007 19:46:21 0800 (PST), bill <b92057@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 on the set {1,2,3,4,5,6,7}. While T is clearly biased in the range 7 to 35, apparently (T mod 7) is _unbiased_ in the range 0 to 6. As to why it's unbiased, I'm not sure. I'll have to think about it.
>Just in case my notation is incorrect; >if T = 21, then T mod 7+1 = 1; >If T = 35, T mod 7 +1 = 1.
Right.
>The following is the typical result of my simulation. > >1 1209 >2 1115 >3 1180 >4 1189 >5 1191 >6 1131 >7 1171 >TOT 8186
Looks uniform enough.
I thought it would be badly biased.
Oh well  at least I was right about the continuous case.
I'm glad you took up the challenge to do a simulation. That's the cool thing about probability. Very often, a simple experiment is all it takes to validate a true claim or invalidate a false one (to within a reasonable doubt).
quasi

