In article <yef16.54659$[EMAIL PROTECTED]>,
Derek Ross <[EMAIL PROTECTED]> wrote:
>This is an odd distribution... I think it may somehow be related to the
>binomial distribution, but I'm not certain.
>The idea is difficult to explain, so here's a "real-life" example of
>generating the distribution:
>I have four coins in my hand. Each coin has a number on both sides. All
>coins have a zero on one side, and the other side has a number from 1 to 4.
>I then hurl the handful of coins at the ground, then add up the values of
>all the coins.
>Here's a list of every possible combination of ways the coins may land, with
>the corresponding total.
<Coins Total
<0 0 0 0 0
<1 0 0 0 1
<0 2 0 0 2
<1 2 0 0 3
<0 0 3 0 3
<1 0 3 0 4
<0 2 3 0 5
<1 2 3 0 6
<0 0 0 4 4
<1 0 0 4 5
<0 2 0 4 6
<1 2 0 4 7
<0 0 3 4 7
<1 0 3 4 8
<0 2 3 4 9
<1 2 3 4 10
>If you plot the histogram, you can see the faint shadow of a bell curve
>there.
>Histograms of totals:
>Total Quantity
> 0 1
> 1 1
> 2 1
> 3 2
> 4 2
> 5 2
> 6 2
> 7 2
> 8 1
> 9 1
> 10 1
>Well then, with all that being said, my question: what kind of distribution
>is this anyway? I know I can calculate everything by hand for a small number
>of coins, but say I had 100 coins, then I would need an equation to generate
>the distribution.
>Any help would be greatly appreciated...
>Derek Ross.
If there is a simple form, it will involve using results
from some version of number theory. The generating function
of the total is (\prod_1^n (1+x^k) )/2^n. If the mean n(n+1)/2
is subtracted, the characteristic function is \prod cos(kt/2).
It is not at all difficult to prove that the sum is asymptotically
normal, as the conditions for the Central Limit Theorem are
satisfied.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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