De Moivre-Laplace and Continuity Correction

The De Moivre-Laplace theorem says that for fixed p in zero to one, a binomial Bin(n,p) is well approximated by a normal distribution with mean np and variance np times (1 minus p) when n is large.

The continuity correction is the half-integer shift that adjusts for the fact that the binomial is discrete and the normal is continuous. For a discrete upper bound k, the corrected approximation uses k plus a half instead of k. The correction is mechanical, costs nothing to apply, and typically improves accuracy by an order of magnitude at the n you see on Exam P.

A portfolio has 100 independent policies, each with probability 0.5 of generating a claim in the year. Estimate the probability of seeing at most 55 claims.

The exact binomial probability is 0.8644. The naive normal approximation uses Phi((55 - 50) / 5) = Phi(1) = 0.8413. The continuity-corrected version uses Phi((55.5 - 50) / 5) = Phi(1.1) = 0.8643. The correction recovers two decimal places of accuracy at zero computational cost.

For Exam P, that is the difference between a correct answer and a wrong-by-one-option answer on multiple choice. The correction is not optional; it is the second-order approximation that lines up the discrete cumulative jump with the continuous normal density.

The standard rule of thumb is np at least 10 and n times (1 minus p) at least 10. Inside that regime the normal approximation with continuity correction is typically accurate to within a fraction of a percentage point in cumulative probabilities.

Outside the regime, especially when p is near zero or near one, the binomial is skewed and the symmetric normal is the wrong shape. In that case use the Poisson approximation if np is moderate, or use the exact binomial if n is small.

Exam P aggregate-count and aggregate-loss problems routinely involve sums of iid indicators or claim counts. Once the sum has mean and variance, the De Moivre-Laplace approximation translates them into a normal probability statement. Without the continuity correction, the answer is systematically biased; with it, the approximation matches the exact binomial CDF to within rounding error at the n levels typical of exam problems.

Beyond Exam P, the same logic appears in capital modeling: a portfolio with many independent risks generating one-or-zero claims gives an aggregate count that is binomial in structure, normally approximable in shape, with the continuity correction recovering accuracy at the precision regulators care about.

Skipping the continuity correction. It is the most common single point loss on Exam P normal-approximation problems. The exam syllabus calls it out by name; the answer choices are calibrated so a student who skips the correction lands on the distractor.

Using the normal approximation when p is very small or very close to one. Even with large n, the binomial is too skewed for the normal to fit. Use the Poisson approximation instead. See the dedicated Poisson approximation page for the rule of thumb.

De Moivre-Laplace is the historical first case of the central limit theorem, proved by De Moivre in 1733 for fair coins and extended by Laplace in 1812 to general p. The general CLT generalizes it from binomial to any iid sum with finite variance. Everything an actuary uses normal-approximation for, from claim-count aggregates to portfolio loss totals, is a downstream application of the same idea.