From Coefficients to Distributions: De~Moivre and the Operational View of Probability

Abstract

We trace a conceptual genealogy from Abraham de Moivre's derivation of the normal curve (1733) to the modern distributional approach to statistics. De Moivre's Approximatio ad Summam Terminorum Binomii gave the first systematic derivation of the Gaussian density, its normalising constant (completed by Stirling's identification of $B = \sqrt{2π}$), and its tail probabilities computed to six decimal places -- more than seventy years before Gauss. His method -- extracting information from probability laws by evaluating sums against indicator probes -- is recognisably an instance of the operational viewpoint that underlies distributional statistics. We identify a four-stage chain: coefficient extraction (De Moivre) $\to$ generating functions (Euler, Laplace) $\to$ characteristic functions (Fourier, Lévy) $\to$ distributional pairings $\langle T, φ\rangle$ (Schwartz). At each stage the probes become more flexible and the class of laws that can be studied grows wider. The distributional framework, in which a probability law is represented by a distribution--kernel pair $(T, φ) \in \mathcal{S}'(\mathbb{R}) \times \mathcal{S}(\mathbb{R})$, is the natural endpoint of this progression. We formulate and prove a distributional version of the De Moivre--Laplace theorem: the standardised binomial distribution converges to the Gaussian in $\mathcal{S}'(\mathbb{R})$, with De Moivre's original computation corresponding to the special case of indicator test functions. We also discuss the transversality framework, which provides a geometric explanation -- via infinite codimension of degeneracy strata -- for why pathologies such as moment indeterminacy, non-identifiability, and singular Fisher information are rarely encountered in parametric statistical models.