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[[File:De moivre-laplace.gif|right|250px|thumb|As ''n'' grows large, the shape of the binomial distribution begins to resemble the smooth [[Gaussian curve]].]]
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In [[probability theory]], the '''de Moivre–Laplace theorem''' is a [[normal distribution|normal]] approximation to the [[binomial distribution]]. It is a special case of the [[central limit theorem]]. It states that the [[binomial distribution]] of the number of "successes" in ''n'' [[statistical independence|independent]] [[Bernoulli trial]]s with probability ''p'' of success on each trial is approximately a [[normal distribution]] with mean ''np'' and standard deviation {{math|{{radical|''np(1-p)''}}}}, if ''n'' is very large and some conditions are satisfied.
 
The theorem appeared in the second edition of ''[[The Doctrine of Chances]]'' by [[Abraham de Moivre]], published in 1738.<!--it appears de Moivre first published it in an article in 1733 or 1734 which later made it into The Doctrine of Chances, but I could't find a reference for that.--> The "Bernoulli trials" were not so-called in that book, but rather de Moivre wrote about the [[probability distribution]] of the number of times "heads" appears when a coin is tossed 3600 times.<ref>{{cite book|chapter=De Moivre on the law of normal probability|last=Walker|first=Helen M|chapterurl=http://www.york.ac.uk/depts/maths/histstat/demoivre.pdf|editor-last=Smith|editor-first=David Eugene|title=A source book in mathematics|year=1985|publisher=Dover|isbn=0-486-64690-4|page=78|quote=But altho’ the taking an infinite number of Experiments be not practicable, yet the preceding Conclusions may very well be applied to finite numbers, provided they be great, for Instance, if 3600 Experiments be taken, make ''n'' {{=}} 3600, hence ½''n'' will be {{=}} 1800, and ½√''n'' 30, then the Probability of the Event’s neither appearing oftner than 1830 times, nor more rarely than 1770, will be 0.682688.}}</ref>
 
== Theorem ==
As ''n'' grows large, for ''k'' in the [[neighborhood (mathematics)|neighborhood]] of ''np'' we can approximate<ref>Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes", 4th Edition</ref><ref>Feller, W. (1968) ''An Introduction to Probability Theory and Its Applications (Volume 1)''. Wiley. ISBN 0-471-25708-7. Section VII.3</ref>
: <math>{n \choose k}\, p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}\,e^{-\frac{(k-np)^2}{2npq}}, \qquad p+q=1,\ p, q > 0</math>
in the sense that the ratio of the left-hand side to the right-hand side converges to 1 as ''n'' → ∞.
 
==Proof==
According to [[Stirling's formula]], we can replace the factorial of a large number ''n'', with the approximation:
 
: <math>n! \simeq  n^n e^{-n}\sqrt{2 \pi n}\qquad \text{as } n \to \infty.</math>
 
Thus
 
: <math>\begin{align}
{n \choose k} p^k q^{n-k} & = \frac{n!}{k!(n-k)!} p^k q^{n-k} \\
& \simeq \frac{n^n e^{-n}\sqrt{2\pi n} }{\left (k^ke^{-k}\sqrt{2\pi k} \right ) \left ((n-k)^{n-k}e^{-(n-k)}\sqrt{2\pi (n-k)} \right )} p^k q^{n-k}\\
& = \left (\frac{\sqrt{2\pi n} }{\sqrt{2\pi k} \sqrt{2\pi (n-k)} }\right ) \cdot \left (\frac{e^{-n}}{e^{-k}e^{-(n-k)} }\right) \cdot \left (\frac{n^n }{k^k(n-k)^{n-k}} p^k q^{n-k}\right )\\
& = \sqrt{\frac{n}{2\pi k(n-k)}}\cdot 1 \cdot \left (n^n\left(\frac{p}{k}\right)^k{\left(\frac{q }{n-k }\right)}^{(n- k)}\right ) \\
& = \sqrt{\frac{n}{2\pi k(n-k)}} \left ( n^{n-k} n^k {\left(\frac{p}{k}\right)}^k {\left(\frac{q}{n-k}\right)}^{(n-k)}\right )\\
& = \sqrt{\frac{n}{2\pi k(n-k)}} \left ( \left(\frac{np }{k }\right)^k \left(\frac{nq }{n-k }\right)^{(n- k)} \right)\\
& = \sqrt{\frac{n}{2\pi k(n-k)}} \left ( \frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq }\right)^{-(n- k)} \\
& = \sqrt{\frac{n}{2\pi k(n-k)}} \left ( 1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)} && x :=\frac{k-np}{\sqrt{npq}} \\
& = \sqrt{\frac{n}{2\pi k(n-k)} \frac{n^{-2}}{n^{-2}} } \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)} \\
& = \sqrt{\frac{n^{-1}}{2\pi k(n-k) n^{-2}}} \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)} \\
& = \sqrt{\frac{n^{-1}}{2\pi \frac{k}{n}\frac{(n-k)}{n}}} \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)}\\
& = \sqrt{\frac{n^{-1}}{2\pi \frac{k}{n}\left(1-\frac{k}{n}\right)}} \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)} \\
&\simeq \sqrt{\frac{n^{-1}}{2\pi p(1-p)}} \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)}  &&\text{as } k\to np \text{ we get } \tfrac{k}{n} \to p\\
& =\sqrt{\frac{1}{2\pi npq}} \left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)} && p+q=1\\
& = \frac{1}{\sqrt{2\pi npq}} \exp \left \{ \ln \left [\left(1+x\sqrt{\frac{q}{np}}\right)^{-k} \left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n - k)}\right ] \right \}&& e^{\ln(y)} = y \\
& = \frac{1}{\sqrt{2\pi npq}} \exp \left \{ \ln \left[\left(1+x\sqrt{\frac{q}{np}}\right)^{-k}\right ]+\ln\left [\left(1-x\sqrt{\frac{p}{nq}}\right)^{-(n-k)}\right ] \right \} \\
& = \frac{1}{\sqrt{2\pi npq}} \exp \left \{ -k\ln\left [1+x\sqrt{\frac{q}{np}} \right ] -(n-k)\ln\left [1-x\sqrt{\frac{p}{nq}}\right ] \right \} \\
& = \frac{1}{\sqrt{2\pi npq}} \exp \left \{  -\left(np+x\sqrt{npq}\right) \ln\left [1+x\sqrt{\frac{q}{np}} \right ] - \left(nq-x\sqrt{npq}\right) \ln\left [1-x\sqrt{\frac{p}{nq}}\right ] \right \} 
\end{align}</math>
 
The last line follows from our definition of ''x''. Now using the Taylor series expansion of the functions ln(1±''x'') we arrive at:
 
:<math>\begin{align}
\frac{1}{\sqrt{2\pi npq}} \exp &\left \{-\left(np+x\sqrt{npq}\right)\left(x\sqrt{\frac{q}{np}}-\frac{x^2q}{2np}+\cdots \right) -\left(nq-x\sqrt{npq}\right)\left(-x\sqrt{\frac{p}{nq}}-\frac{x^2p}{2nq}-\cdots \right)\right \}  = \\
& =\frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\left(x\sqrt{npq}-\tfrac{1}{2}x^2q+x^2q+\cdots \right)-\left(-x\sqrt{npq}-\tfrac{1}{2}x^2p+x^2p+\cdots \right) \right \}\\
& =\frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\left(x\sqrt{npq}+\tfrac{1}{2}x^2q+\cdots \right)-\left(-x\sqrt{npq}+\tfrac{1}{2}x^2p+\cdots \right) \right \}\\
& =\frac{1}{\sqrt{2\pi npq}} \exp \left \{ -x\sqrt{npq}-\tfrac{1}{2}x^2q+x\sqrt{npq}-\tfrac{1}{2}x^2p-\cdots \right \} \\
& =\frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\tfrac{1}{2}x^2\left(q+p\right)-\cdots \right \} \\
& =\frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\tfrac{1}{2}x^2-\cdots \right \}\\
&\simeq \frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\tfrac{1}{2}x^2 \right \} && \text{as } n \to \infty  \text{ we get } x \to 0\\
&= \frac{1}{\sqrt{2\pi npq}} \exp \left \{ -\frac{1}{2} \left(\frac{k-np}{\sqrt{npq}}\right)^2 \right \} \\
&= \frac{1}{\sqrt{2\pi npq}} \exp \left \{ - \frac{(k-np)^2}{2npq} \right \}
\end{align}</math>
 
Thus,
 
: <math>{n \choose k}p^kq^{n-k}\simeq \frac{1}{\sqrt{2\pi npq}}e^{-\frac{(k-np)^2}{2npq}}.</math>
 
== Notes ==
{{reflist}}
 
{{DEFAULTSORT:De Moivre-Laplace, Theorem Of}}
[[Category:Central limit theorem]]

Latest revision as of 20:35, 7 November 2014

Pay sites require associates to subscribe in order to communicate with other members. Costs differ, but generally, reduce month-to-month prices are available to those who subscribe for longer intervals. Initially, nevertheless, you may favor to spend the higher single-thirty day period cost in order to get a really feel for the services and for this courting format.

There's no difference in attempting to satisfy a companion online as there is offline. It's just like shopping. Some people do it offline, and now hundreds of thousands of individuals are performing it online. People alter with the occasions. There's no purpose not to place a totally free profile on a few nerd dating services to see which one will experience the very best benefits. Someone stepping out of their ease and comfort zone frequently has numerous much more advantages than staying in their ease and comfort zone.

Keep in mind; your discussions have to flow naturally. If it feels like either of you are pulling teeth to keep it heading then there will be some problems.



This might be why singles are using background checks to investigate the background of their possible lover. Contemplating the compressed time for meeting and evaluating, this seems like the smart factor to do. Or else, how do you know who you are truly assembly on line? Are they really who they say they are? If you are a single parent and courting online or through nerd dating (read this article) services or companies, you experienced much better be sure Mr. Wonderful doesn't prefer your children over you.

If you are a newbie when it comes to online dating service, then for starters try making your profile and then begin interacting with a bunch of people. You can introduce yourself in the initial message, but don't go overboard and create some overtly lengthy biography about your self! Maintain it short and simple, you do not want to scare them absent with your preliminary message. Keep in mind your etiquette as well! Although you might be anonymous but you still have to have some sort of internet etiquette otherwise you are heading to be ignored or even blocked!

Even if your profile is excellent, many people will query why you have not posted a image. They will start to make speculations, bad judgments, or remove you from their checklist of prospects. And, of program, you do not want this kind of things to occur.

To evaluate quality and worth means you have to spend interest. You have to research issues. But, clearly, when we are busy with your digital distractions we don't really pay interest. It is like the people we all know who journey to some unique locale rich with architectural and cultural wonders, and the best they can do is tell you about the food.

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By signing up on any of these websites, you will have more to keep you active drinking your early morning espresso than four pages of comics, the enjoyment pages, and YouTube. Sifting through possibly interesting individuals can open up the doorway to a 5 moment meeting, a fifteen moment phone discussion, or even a fantastic boyfriend. And really, this can all take place in the comfort of your home or via your mobile telephone if you so want. You can spend as much or as little time doing this as you wish.

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