De Moivre–Laplace theorem
- De Moivre–Laplace theorem
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 , if "n" is very large and some conditions are satisfied.
The theorem first appeared in "The Doctrine of Chances" by Abraham de Moivre, published in 1733. 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.Fact|date=September 2008
The theorem
If , then for "k" in the -neighborhood of "np", we can approximate [Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes", 4th Edition]
:
The limiting form of theorem states that
:
as
References
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