De Moivre–Laplace theorem

In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particular, the theorem shows that the probability mass function of the random number of "successes" observed in a series of n {\displaystyle n} independent Bernoulli trials, each having probability p {\displaystyle p} of success (a binomial distribution with n {\displaystyle n} trials), converges to the probability density function of the normal distribution with expectation n p {\displaystyle np} and standard deviation n p ( 1 − p ) {\textstyle {\sqrt {np(1-p)}}} , as n {\displaystyle n} grows large, assuming p {\displaystyle p} is not 0 {\displaystyle 0} or 1 {\displaystyle 1} .

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