Wikipedia:Reference desk/Archives/Mathematics/2008 December 19

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December 19

Probability regularisation

Is there any concept in Mathematics regarding regularisation of outcomes with an increase in the number of trials; e.g if I toss a coin a thousand times, that there will be a better chance of me getting nearly equal number of heads and tails, rather than if I were to toss the coin 100 times (and further such regularisations for any event)?Leif edling (talk) 09:41, 19 December 2008 (UTC)

"Regularisation" principles in statistics and probability include the law of large numbers and the central limit theorem. For an example of a regularisation principle which does not hold, see gambler's fallacy. Gandalf61 (talk) 11:26, 19 December 2008 (UTC)

That's a classic problem treated in the first half of the 18th century in Abraham de Moivre's book The Doctrine of Chances (and of course also in the articles titled central limit theorem, continuity correction, and law of large numbers). Michael Hardy (talk) 23:46, 22 December 2008 (UTC)

Boubaker polynomials

Dear WP user (interested in Maths)

I am K. Duvvuri from Indian Univesity, I am working on Special Functions.

Please can you help me (help or hints) concerning two items:

- the existence of an exponential generating fonction to the Boubaker polynomials (see page :http://en.wikipedia.org/wiki/Boubaker_polynomials)

- the eventual ANALYTICAL expression of the reoots of these polynomials. Please answer here, in my talk page or in my email: --------

Thank you for help. Duvvuri.kapur (talk) 10:12, 19 December 2008 (UTC)

If you look at the article on Boubaker polynomials, you will find the expression for their ordinary generating function; given the ordinary generating function it is very easy to find the exponential generating function, follow those links for more information. Unfortunately I can't help with finding the polynomial roots off-hand. Eric. 68.18.63.75 (talk) 18:48, 19 December 2008 (UTC)

Whoops, I got my definitions mixed up. Eric. 68.18.63.75 (talk) 18:54, 19 December 2008 (UTC)

The exponential generating function can be found from a recursive definition in a way analogous to the ordinary generating function, but using partial differential with respect to t instead of multiplication by t. Suppose $B_{n}(x)=xB_{n-1}(x)-B_{n-2}(x)$; let

$f(x,t)=\sum _{n\geq 0}B_{n}(x){\frac {t^{n}}{n!}}$

Then, where f' denotes the first derivative of f with respect to t, and f' ' the second derivative, we have

$f''-xf'+f=0$

So, if $\lambda _{i}$ are the solutions to $\lambda ^{2}-x\lambda +1=0$, then the solutions to the above differential equation are (except when $x=\pm 2$)

$f(x,t)=Ae^{\lambda _{1}t}+Ce^{\lambda _{2}t}$.

From the initial conditions $B_{0}(x)=1$ and $B_{1}(x)=x$ we find A + C = 1 and $A\lambda _{1}+C\lambda _{2}=x$. This identifies a unique solution f, which is our exponential generating function.

The trouble is, that after going through this I realized that the recursive equation only holds for n > 2, not for n = 2. So some simple changes need to be made to adjust for that, but I believe the general technique should still work. Eric. 68.18.63.75 (talk) 21:56, 19 December 2008 (UTC)