de Moivre–Laplace theorem

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In probability theory, the de Moivre–Laplace theorem is a 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 independent Bernoulli trials with probability p of success on each trial is approximately a normal distribution with mean np and standard deviation \sqrt{npq}, 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. 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 1800 times.[citation needed]

[edit] The theorem

If n \rightarrow \infty, then for k in the \sqrt{npq}-neighborhood of np, we can approximate[1][2]

\left( \begin{array}{c} n \\ k \end{array} \right) p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / 2npq}, \ \ p+q=1, \ p>0, \ q>0.

The limiting form of the theorem states that[1][2]

\frac{\sqrt{2 \pi npq} \left( \begin{array}{c} n \\ k \end{array} \right) p^k q^{n-k}}{e^{-(k-np)^2 / 2npq}} \rightarrow 1

as n \rightarrow \infty.

[edit] Notes

  1. ^ a b Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes", 4th Edition
  2. ^ a b Feller, W. (1968) An Introduction to Probability Theory and Its Applications (Volume 1). Wiley. ISBN 0-471-25708-7. Section VII.3
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