ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Normal approximation to binomial probabilities

Example

Assume AC and fix $0 < p < 1$ . Write $Φ (x) = \int_{- \infty}^{x} e^{- y^{2} / 2} / \sqrt{2 π} d y$ . For fixed real $a < b$ and $B_{n} \sim Bin (n, p)$ for every integer $n \geq 1$ , $P ⁣ (a \leq \frac{B_{n} - n p}{\sqrt{n p (1 - p)}} \leq b) ⟶ Φ (b) - Φ (a) .$ For a finite integer interval, continuity correction is a numerical approximation, not an error bound supplied by this theorem.

Facts & Assumptions

[F1]

Under AC standardized binomial laws converge to N(0,1). De Moivre-Laplace central limit theorem.

[F2]

Weak convergence gives probabilities of target continuity sets. Portmanteau theorem.

[F3]

The standard normal has density exp(-x^2/2)/sqrt(2pi). Standard normal and normal laws.

[F4]

Finite endpoint sets are Lebesgue null under countable choice. Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ .

Verification

Given: Assume AC and fix $0 < p < 1$ . Write $Φ (x) = \int_{- \infty}^{x} e^{- y^{2} / 2} / \sqrt{2 π} d y$ . For fixed real $a < b$ and $B_{n} \sim Bin (n, p)$ for every integer $n \geq 1$ , $P ⁣ (a \leq \frac{B_{n} - n p}{\sqrt{n p (1 - p)}} \leq b) ⟶ Φ (b) - Φ (a) .$ For a finite integer interval, continuity correction is a numerical approximation, not an error bound supplied by this theorem.

1.1

The normal law assigns zero mass to each singleton: its bounded density integrates to zero on a Lebesgue-null singleton by [F3]–[F4]. The boundary of [a,b] is contained in the two endpoints, so it is a continuity set. Apply [F1] and the continuity-set implication of [F2] to get the displayed limit. Open, closed or half-open choices of the two fixed standardized endpoints have the same limit. If a=b the closed singleton has limiting probability zero; if a>b the event is empty.

F1F2F3F4

2.1

Take n=100,p=1/2 and the event $45 \leq B_{100} \leq 55$ . Its mean is 50 and standard deviation is $\sqrt{100 / 4} = 5$ . Raw standardization gives endpoints -1 and 1, and the corresponding normal probability is $Φ (1) - Φ (- 1) \approx 0.68268949$ . The half-unit cell endpoints $89 / 2$ and $111 / 2$ give corrected standardized endpoints -1.1 and 1.1, and the corrected normal probability is $Φ (1.1) - Φ (- 1.1) \approx 0.72866788$ . These decimal evaluations are of the displayed normal integrals. The theorem does not bound either finite-n approximation error or prove that the correction always improves it. AC is inherited through [F1] and the normal-density and null-set construction.

step 1.1F1F3∎

Depends on

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Sources

Durrett, Probability: Theory and Examples, Section 3.1 (standard reference, not scraped)
Aldous and Chewi, Probability Theory notes, Corollary 6.1 (standard reference, not scraped)