Re: Birthday problem for such non-uniform birthday probabilities


"quasi" <quasi@xxxxxxxx> wrote in message
news:5604o1h2nfn6am6o1fvqr80slg5qmnccgh@xxxxxxxxxx
> On Mon, 21 Nov 2005 10:30:59 -0600, "Doug" <nospam@xxxxxxxxxx> wrote:
>
>>
>>"David M Einstein" <Deinst@xxxxxxxxx> wrote in message
>>news:1132586540.046923.115340@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>>>
>>> Doug wrote:
>>>> "Helmut Zeisel" <helmut.zeisel@xxxxxx> wrote in message
>>>> news:dls5fv.5lg.1@xxxxxxxxxxxxxxxxx
>>>> > Consider the birthday problem:
>>>> >
>>>> > There are n randomly chosen person in a room. What is the
>>>> > proabability
>>>> > that there exist k persons who have birthday on the same day.
>>>> >
>>>> > I know how to compute the probability assuming a uniform birthday
>>>> > distribution.
>>>> >
>>>> > It is reasonable that this probability increases for a non-uniform
>>>> > birthday distribution.
>>>> >
>>>> > Where can I find a proof for this result?
>>>
>>> Blom, D. (1973), "A birthday problem", American Mathematical Monthly,
>>> vol. 80, pp. 1141-1142
>>>
>>>> >
>>>> > Helmut
>>>>
>>>> You won't, as it is also reasonable that this probability decreases for
>>>> a
>>>> non-uniform
>>>> distribution.
>>>>
>>>> You need to specify the "non-uniform" distributions to determine if it
>>>> increases/decreases.
>>>
>>> I would be very interested in seeing a distribution that decreases the
>>> probability of
>>> a match.
>>>
>>
>>simple- define "distribution" as sequential days beginning on Jan 1
>>
>
> Wrong.
>
> Remember, no matter what distribution you specify, we are selecting
> randomly from it, so if k>1, repetitions are always possible.
>
> If the distribution is not uniform, the probability of a match in k
> sample elements, where n>=k>1, is always greater than it would be if
> the distribution was uniform,
>
> It's not hard to prove this in general, but as an example, assume a 2
> day year and let n>=k=2.
>
> Number the days of the year 1,2.
>
> (Note that for this type of year, every day is either New Year's Day
> or New Year's Eve)
>
> Let p1, p2 be the probabilities of having a birthday on days 1,2
> respectively.
>
> Since n>=k=2, the probability of a match can be calculated exactly as
> p1^2+p2^2.
>
> Since p1, p2 are nonnegative,
>
> p1^2+p2^2 >= 2*p1*p2 with equality iff p1=p2.
>
> Then, adding p1^2+p2^2 to both sides and factoring, we get
>
> 2(p1^2+p2^2) >= (p1+p2)^2
>
> Since p1+p2=1, we get
>
> p1^2+p2^2 >= 1/2 with equality iff p1=p2.
>
> Hence if the distribution is uniform, the probability of a match is
> exactly 1/2, otherwise it's strictly greater than 1/2.
>
> quasi

nice show, but still wrong.

You rely on a "continuous" non-uniform distribution, not a non-uniform
discrete distribution, or a distribution that is dependent upon previous
selections, like removing a ball at a time from a container.

Leap year is less.

Assume a 1000 day year. Also far less than original problem.

a good learning lesson for you.


.

Relevant Pages

  • Re: Pigeons, People, and Priors
    ... the variance of the probability generator go to zero you have a continuum ... a random-interval 60 s schedule is not. ... The Exponential Distribution ... I probably should have used the phrase "statistical learning theory" rather ...
    (comp.ai.philosophy)
  • Re: Cantorian pseudomathematics
    ... >> I understand the probability a divides n when we have a uniform ... >> But there is no uniform distribution on N, so I don't see what Han ... no way to do that with a countable set of discrete ...
    (sci.math)
  • Re: So called "stimulus/response" models
    ... Instead of answering to each misunderstood, ironic and out of context ... Sorry, you exhibit a simplistic view of probability theory, and an even more ... of acquiring the consequences of responses. ... distribution over consequences of a given act. ...
    (comp.ai.philosophy)
  • Re: Calculus XOR Probability
    ... case where you have a uniform probability distribution over an infinite set of ... Um, I'm not a probability expert, but isn't it the case that given a ... distribution, you can say what the _expected value_ of a random ... uniform distribution will be recognisably ordinary pofnats, ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... Han is talking about the probability that a randomly chosen ... >> segment of the naturals, and the limit of this probability as that initial ... > But there is no uniform distribution on N, so I don't see what Han ...
    (sci.math)
Quantcast