Talk:De Moivre–Laplace theorem

Speed of convergence?

Do we know anything about the speed of convergence? 193.224.40.254 (talk) 10:03, 1 October 2010 (UTC) Sorry, it was Koczy (talk) 10:04, 1 October 2010 (UTC)

Check the central limit theorem and Berry–Esseen theorem articles.  // stpasha »  04:02, 2 October 2010 (UTC)

Proofs and Wikipedia

Does it really make sense to put a proof into Wikipedia?

It should be sufficient to mention the relevant theorems and point to literature.

--Chire (talk) 13:39, 18 September 2013 (UTC)

no, don't touch the proof. — Preceding unsigned comment added by 86.201.147.141 (talk) 08:51, 9 April 2014 (UTC)

The proof has some gaps. For example:

- The negligibility of the elided terms of order 3 or more
- The applicability of the series for log

It is better to give a correct proof or at least indicate that this is a sketch which needs some further (not entirely trivial) work to complete. — Preceding unsigned comment added by 14.139.227.196 (talk) 16:56, 15 February 2016 (UTC)

Making the proof more accessible.

The purpose of my edits is to make this proof more accessible to high school students who have taken calculus. Most students now learn the central limit theorem but the proof in its most general form is beyond them. This proof can be somewhat better explained and more easily understood by the interested student and teacher.

The changes are simply to break the existing proof into parts and reorder the steps a bit to make it clear that the proof consists of successively applying three approximations. An important detail that has been overlooked is also dealt with at the start of the proof.

Boppajim (talk) 02:03, 26 June 2015 (UTC)

The proof should show LHS/RHS approaches 1

Just showing limit of LHS = RHS is not useful. For example, $\lim _{x\to \infty }1/x^{2}=1/x$. This tells us nothing. We should really show $\lim _{n\to \infty ,k\to \infty }{\frac {LHS}{RHS}}=1$. This is non-trivial, though, in this case. — Preceding unsigned comment added by 153.18.72.65 (talk) 18:10, 16 November 2017 (UTC)

The article does not show that the limits are equal. It shows that the two functions of n are asymptotically equivalent, which is precisely the same as saying their ratio has a limit of 1. I'm not saying the proof cannot be made more clear, but your objection as stated is simply not a valid one. (In your example, the two functions of x are not asymptotically equivalent.) - dcljr (talk) 06:04, 17 November 2017 (UTC)

It is unclear to me asymptotic equivalence is preserved, which is indeed the same as saying the ratio has a limit of 1, for the various approximations in the derivation. As the final answer is correct, I am sure asymptotic equivalence is preserved, but that is nowhere near obvious. The way to prove such equivalence is to take ratios and show their limit is 1. 50.242.69.57 (talk)

I think the only realistic option is to simply state that each "$\simeq$" in the proof is, in fact, a statement about asymptotic equivalence. I've added such a statement at the end of the proof. The proof is already too detailed as it is; we don't need to show/justify every step as if we're working out a homework problem. - dcljr (talk) 06:49, 18 November 2017 (UTC)

A new simpler proof in the CMJ

The proof is mine but as this is a math proof I guess there is no conflict of interest. CMJ is the most prestigious journal for this kind of a proof.

Proof

The theorem can be more rigorously stated as follows: $\left(X\!\,-\!\,np\right)\!/\!{\sqrt {npq}}$, with $\textstyle X$ a binomially distributed random variable, approaches the standard normal as $n\!\to \!\infty$, with the ratio of the probabiity mass of $X$ to the limiting normal density being 1. This can be shown for an arbitrary nonzero and finite point $c$. On the unscaled curve for $X$, this would be a point $k$ given by

$k=np-c{\sqrt {npq}}$

For example, with $c$ at 3, $k$ stays 3 sd from the mean in the unscaled curve.

The normal distribution is defined by the differential equation (DE)

$f'\!(x)\!=\!-\!\,{\frac {x-\mu }{\sigma ^{2}}}f(x)$ with initial condition set by the probability axiom $\int _{-\infty }^{\infty }\!f(x)\,dx\!=\!1$.

The binomial distribution limit approaches the normal if the binomial satisfies this DE. As the binomial is discrete the equation starts as a difference equation whose limit morphs to a DE. Difference equations use the discrete derivative, $\textstyle p(k\!+\!1)\!-\!p(k)$, the change for step size 1. As $\textstyle n\!\to \!\infty$, the discrete derivative becomes the continuous derivative. Hence the proof need show only that, for the unscaled binomial distribution,

${\frac {f'\!(x)}{f\!(x)}}\!\cdot \!-\!\,{\frac {\sigma ^{2}}{x-\mu }}\!\to \!1$ as $n\!\to \!\infty$.

The required result can be shown directly:

${\begin{aligned}{\frac {f'\!(x)}{f\!(x)}}{\frac {npq}{np\!\,-\!\,k}}\!&={\frac {p\left(n,k+1\right)-p\left(n,k\right)}{p\left(n,k\right)}}{\frac {\sqrt {npq}}{c}}\\&={\frac {np-k-q}{kq+q}}{\frac {\sqrt {npq}}{c}}\\&={\frac {c{\sqrt {npq}}-q}{npq-cq{\sqrt {npq}}+q}}{\frac {\sqrt {npq}}{c}}\\&\to 1\end{aligned}}$

The last holds because the term $cnpq$ dominates both the denominator and the numerator as $n\!\to \!\infty$.

As $\textstyle k$ takes just integral values, the constant $\textstyle c$ is subject to a rounding error. However, the maximum of this error, $\textstyle {0.5}/\!{\sqrt {npq}}$, is a vanishing value.

The article is available here: http://www.employees.org/~ajoyk/DeMoivre-LaplaceTheorem.pdf — Preceding unsigned comment added by 171.64.66.201 (talk) 01:17, 17 April 2018 (UTC)

I'm sorry, but I think this proof does not work. In your third to last line if we substitute $c={\frac {np-k}{\sqrt {npq}}}$ in the expression ${\frac {np-k-q}{kq+q}}{\frac {\sqrt {npq}}{c}}$, we obtain the quantity ${\frac {np-k-q}{\left(np-k\right)\left(kq+q\right)}}\left(npq\right)$. It is apparent that, for fixed $k,p$, this quantity diverges to $+\infty$ as $n\to +\infty$. Moreover, even in the case it had converged to $1$, I do not think it is clear what this would say about the actual De Moivre theorem; there should at least be a missing passage linking the result (not) obtained with the desired $\mathrm {lim} _{n\to +\infty }{\frac {p(n,k)}{{\frac {1}{\sqrt {2\pi npq}}}e^{-{\frac {(k-np)^{2}}{2npq}}}}}=1.$ 176.206.14.177 (talk) 16:06, 2 February 2024 (UTC)

References