Probability Theory 32 | De Moivre–Laplace theorem

The De Moivre–Laplace theorem turns the binomial distribution into an (asymptotically) normal one, giving a practical way to approximate binomial probabilities with the familiar bell curve from the central limit theorem. The key idea is that when a binomial random variable comes from many independent Bernoulli trials, its distribution becomes increasingly bell-shaped as the number of trials grows—especially near its mean.

A binomial distribution counts the number of “successes” in n repeated Bernoulli (coin-toss) experiments, where each trial succeeds with probability p and fails with probability 1−p. In the transcript’s coin-toss picture, each bead falling into a column corresponds to a particular count K of successes, with K ranging from 0 to n. The exact probability mass function is

P(X=K)=\binom{n}{K} p^K (1−p)^{n−K}.

For small n, the distribution is discrete and can be computed directly. But as n increases, histograms built from repeated simulations start to resemble a continuous bell curve, motivating the approximation: instead of wrestling with binomial coefficients, one can use the normal distribution’s density.

The normal approximation uses the mean and variance of the binomial: mean n·p and variance n·p(1−p). The transcript presents the approximate probability (for K near the middle) as matching the normal density up to the correct scaling factor. Concretely, the approximation takes the form

P(X=K) ≈ \frac{1}{\sqrt{2\pi n p(1−p)}}\exp\left(-\frac{(K−n p)^2}{2 n p(1−p)}\right).

This is the practical De Moivre–Laplace message: for large n and K close to the expected value n·p, binomial probabilities behave like normal probabilities.

To move beyond a hand-wavy match of shapes, the transcript describes a more precise limiting statement. The normal approximation is justified by comparing the binomial pmf to the normal pdf (with the appropriate standardization). The ratio of the two quantities approaches 1 as n→∞, meaning the difference between the discrete binomial probability and the continuous normal density becomes negligible in the relevant region.

That “relevant region” matters: K must stay near the mean. In the limit formulation, the exponent term in the normal expression must remain bounded—equivalently, (K−n p)^2 is controlled relative to n p(1−p). Under these conditions, after shifting and scaling (the same standardization used in the central limit theorem), the binomial distribution converges pointwise to the standard bell curve.

Finally, the theorem’s assumptions exclude the degenerate endpoints p=0 and p=1, requiring 0<p<1. With that, the De Moivre–Laplace theorem becomes a named, historically important special case of the central limit theorem: it formalizes when and how binomial probabilities can be replaced by normal ones for computation and approximation.