Approximating a Binomial Distribution | 二项分布的近似

📚 Approximating a Binomial Distribution | 二项分布的近似

In Edexcel A Level Mathematics, you often meet binomial probabilities that would be extremely tedious to compute using the formula P(X = r) = ⁿCᵣ pʳ(1-p)ⁿ⁻ʳ. When n is large, a binomial distribution can be approximated by a Poisson distribution or a normal distribution. This article explains the conditions, formulas, continuity corrections and common exam pitfalls.

在 Edexcel A Level 数学中，我们经常会遇到用公式 P(X = r) = ⁿCᵣ pʳ(1-p)ⁿ⁻ʳ 计算起来非常繁琐的二项概率。当 n 很大时，二项分布可以用泊松分布或正态分布来近似。本文将解释近似条件、公式、连续性校正以及常见考试易错点。

---

1. Why Approximate a Binomial Distribution? | 为什么近似二项分布？

A binomial random variable X ~ B(n, p) counts the number of successes in n independent trials, each with probability p of success. Its exact probability function is P(X = r) = ⁿCᵣ pʳ(1-p)ⁿ⁻ʳ.

二项随机变量 X ~ B(n, p) 表示 n 次独立试验中成功的次数，每次成功概率为 p。其精确概率函数为 P(X = r) = ⁿCᵣ pʳ(1-p)ⁿ⁻ʳ。

For very large n, calculating ⁿCᵣ and powers such as pʳ and (1-p)ⁿ⁻ʳ by hand is not practical. Approximations give a fast and reliable way to estimate probabilities, especially in hypothesis tests and real-world modelling.

当 n 非常大时，手工计算 ⁿCᵣ 以及 pʳ、(1-p)ⁿ⁻ʳ 的幂并不现实。近似方法提供了一种快速可靠的概率估计方式，尤其在假设检验和实际建模中非常有用。

There are two principal approximations: Poisson, used when p is very small, and normal, used when p is not too close to 0 or 1 and n is large.

主要有两种近似：当 p 非常小时使用泊松近似；当 p 不太接近 0 或 1 且 n 较大时使用正态近似。

---

2. Key Parameters of a Binomial Distribution | 二项分布的关键参数

For X ~ B(n, p), the mean and variance are:

对于 X ~ B(n, p)，均值与方差为：

E(X) = np, Var(X) = np(1-p)

These two parameters control every approximation. The Poisson approximation matches the mean λ = np. The normal approximation matches both the mean np and the variance np(1-p).

这两个参数决定了所有近似方法。泊松近似匹配均值 λ = np；正态近似同时匹配均值 np 与方差 np(1-p)。

Let q = 1 – p. The variance is often written as npq. In normal approximation conditions, both np and nq must be sufficiently large.

令 q = 1 – p，方差通常写作 npq。在正态近似条件中，np 和 nq 都必须足够大。

---

3. Poisson Approximation: Conditions | 泊松近似的条件

If X ~ B(n, p), and n is large while p is small, then X can be approximated by a Poisson distribution with parameter λ = np.

若 X ~ B(n, p)，当 n 很大且 p 很小时，X 可以用参数 λ = np 的泊松分布来近似。

Typical rule of thumb used in Edexcel questions:

Edexcel 题目中常用的经验法则：

• n is large, usually n > 50
• λ = np < 5, or p < 0.1

In some contexts, np < 5 is the main check; in others, p < 0.1 with large n is enough. Always look for ‘large n, small p’ in the question.

在某些情况下，np < 5 是主要判断标准；在另一些情况下，p < 0.1 且 n 足够大即可。做题时要注意题目中 ‘大 n、小 p’ 的特征。

The Poisson approximation works because when n is large and p is small, the binomial distribution is heavily skewed to the right and the event is rare.

泊松近似之所以有效，是因为当 n 很大且 p 很小时，二项分布严重右偏，事件非常稀有。

---

4. Poisson Approximation: Formula and Example | 泊松近似公式与例题

If X ~ B(n, p) is approximated by Y ~ Po(λ), where λ = np, then:

若 X ~ B(n, p) 近似为 Y ~ Po(λ)，其中 λ = np，则：

P(X = r) ≈ e⁻λ λʳ ÷ r!

This avoids binomial coefficients entirely. You only need a calculator value for e⁻λ and powers λʳ.

这完全避免了对二项系数的计算。你只需要用计算器求出 e⁻λ 和 λʳ 的幂。

Example: A factory produces components, and 2% are defective. A sample of 100 components is taken. Find the probability that exactly 3 components are defective.

例题：某工厂生产的元件中有 2% 为次品。随机抽取 100 个元件，求恰好有 3 个次品的概率。

Here X ~ B(100, 0.02). Since n = 100 is large and p = 0.02 is small, use Poisson with λ = np = 100 × 0.02 = 2.

这里 X ~ B(100, 0.02)。因为 n = 100 较大且 p = 0.02 较小，使用 λ = np = 100 × 0.02 = 2 的泊松近似。

P(X = 3) ≈ e⁻² × 2³ ÷ 3! = 0.1804

The exact binomial probability is 0.1823, so the Poisson approximation is very close.

精确的二项概率为 0.1823，因此泊松近似非常接近。

---

5. Normal Approximation: Conditions | 正态近似的条件

If X ~ B(n, p), and n is large with p not too close to 0 or 1, then X

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）