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


Re: Random numbers
Posted:
Dec 23, 2007 9:33 PM


On Sun, 23 Dec 2007 16:06:30 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. >> >> >If RN_7 = T/7, the probability >> >of a correct guess is < .11 if you always >> >guess that T = 21 or 22 >> >> If the original RNG is uniformly distributed on the interval (1,5), >> then it's a continuous distribution, so the probability that T = 21 or >> T = 22 is 0. > >That makes it even harder to guess correctly if you have to be exact. > >> And once again, since T/7 only has range 1 to 5, thus it's obviously >> not uniform on (1,7). It's not even uniform on (1,5), since it has >> more concentration near the mean (3) than near the ends. >> >> quasi > >Assume that RNG_5 is continuous in the interval 1  5 >and 0 elsewhere. Let X_i be a number generated by RNG_5. >Then Y_i = 1.5 * (X_i)  0.5 is continuous in the interval 1  7 >and 0 elsewhere.
Yes, and Y is uniform if X is.
The problem with your previously proposed RNGs was the dependence on the variable T. But T is not even close to uniform, being biased towards its mean, and away from the ends. Of the RNGs proposed in this thread, all of those which were based on T were biased, and badly so.
>That is, unless there are more points between 1 and 7 than there are >between 1 and 5?
Well, that's the "gaps" issue, relating to the fact that a "continuous" _pseudorandom_ number generator is actually only _pretending_ to be continuous. In reality, it's discrete. But if we are just discussing theoretical random number generators, that issue doesn't come into play.
quasi

