Poisson Approximation to Binomial Distribution: Conditions and Methods | 泊松分布近似二项分布的条件与方法

📚 Poisson Approximation to Binomial Distribution: Conditions and Methods | 泊松分布近似二项分布的条件与方法

The binomial distribution is a fundamental model for counting successes in a fixed number of independent trials. When the number of trials n is very large and the probability of success p is very small, exact binomial calculations can become tedious. The Poisson distribution provides a powerful and elegant approximation in these situations, reducing computation while giving accurate results within well-defined conditions.

二项分布是描述固定次数独立试验中成功次数的基本模型。当试验次数 n 很大且成功概率 p 很小时，精确计算二项分布概率往往十分繁琐。泊松分布在这种情况下提供了一种强大而简洁的近似方法，既能减少计算量，又能在明确的条件内给出准确结果。

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1. Revisiting the Binomial Distribution | 回顾二项分布

If a random variable X follows a binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of observing exactly x successes is:

P(X = x) = ⁿCₓ pˣ(1 − p)ⁿ⁻ˣ

The mean and variance of a binomial distribution are:

E(X) = np,   Var(X) = np(1 − p)

These formulas are exact, but when n reaches hundreds or thousands, computing combinations such as ²⁰⁰C₅ by hand becomes extremely time-consuming.

若随机变量 X 服从参数为 n 和 p 的二项分布，记作 X ~ B(n, p)。恰好出现 x 次成功的概率为：

P(X = x) = ⁿCₓ pˣ(1 − p)ⁿ⁻ˣ

二项分布的均值和方差分别为：

E(X) = np，  Var(X) = np(1 − p)

这些公式是精确的，但当 n 达到数百或数千时，手工计算 ²⁰⁰C₅ 这样的组合数会非常耗时。

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2. The Poisson Distribution as a Limit | 泊松分布作为极限

The Poisson distribution arises naturally as the limiting form of the binomial distribution when n tends to infinity and p tends to zero, while the product np remains equal to a positive constant λ. This is the key idea behind the approximation.

泊松分布是二项分布的极限形式：当 n 趋于无穷大、p 趋于零，并且乘积 np 保持为正常数 λ 时，二项分布就趋向于泊松分布。这正是近似的核心思想。

Mathematically, for every fixed x = 0, 1, 2, …:

lim_n→∞ ⁿCₓ pˣ(1 − p)ⁿ⁻ˣ = e^−λλˣ / x!

Therefore, when n is large and p is small, the binomial probabilities are close to those of a Poisson distribution with parameter λ = np.

从数学上看，对每个固定的 x = 0, 1, 2, …，有：

lim_n→∞ ⁿCₓ pˣ(1 − p)ⁿ⁻ˣ = e^−λλˣ / x!

因此，当 n 很大、p 很小时，二项分布概率会非常接近参数 λ = np 的泊松分布。

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3. Conditions for a Valid Approximation | 有效近似的条件

To use the Poisson approximation safely in an exam, you must check that the following conditions hold:

• The number of trials n is large: usually n ≥ 50.

  试验次数 n 要足够大：通常取 n ≥ 50。

• The probability of success p is small: usually p ≤ 0.1.

  成功概率 p 要足够小：通常取 p ≤ 0.1。

• The mean λ = np is moderate, commonly λ ≤ 5. If λ is too large, the Poisson shape becomes symmetric and a normal approximation may be better.

  均值 λ = np 应适中，一般 λ ≤ 5。如果 λ 过大，泊松分布会趋于对称，此时正态近似可能更合适。

• The trials must be independent and have the same probability of success, as required for the binomial model itself.

  各次试验必须独立且成功概率相同，这是二项模型本身的前提。

These conditions ensure that the approximation error is acceptably small for most examination purposes. The approximation improves as n increases and p decreases.

这些条件确保近似误差在考试可接受的范围内。近似效果随着 n 增大和 p 减小而变得越来越好。

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4. Choosing λ: The Mean Must Match | 选择参数 λ：均值要一致

The most natural way to choose the Poisson parameter is to match the mean of the two distributions. Since the mean of B(n, p) is np, we set:

λ = np

This is not an arbitrary choice. The Poisson distribution with parameter λ has mean λ and variance λ. For the binomial distribution, the variance is np(1 − p). When p is small, 1 − p is close to 1, so the binomial variance is approximately np, matching the Poisson variance as well.

选择泊松参数最自然的方法是使两个分布的均值相等。二项分布 B(n, p) 的均值为 np，因此令：

λ = np

这并非随意选择。参数为 λ 的泊松分布均值为 λ，方差也为 λ。二项分布的方差为 np(1 − p)；当 p 很小时，1 − p 接近 1，所以二项方差约为 np，与泊松方差也相匹配。

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5. Step-by-Step Method | 分步方法

Follow these steps when applying the Poisson approximation to an exam problem:

在考试中运用泊松近似时，请遵循以下步骤：

• Step 1: Identify that X is binomial, X ~ B(n, p), and state the conditions.

  第一步：确认 X 服从二项分布 X ~ B(n, p)，并说明条件。

• Step 2: Check that n ≥ 50, p ≤ 0.1, and λ = np ≤ 5 (or another specified threshold).

  第二步：检查是否满足 n ≥ 50，p ≤ 0.1，且 λ = np ≤ 5（或题目给定的其他标准）。

• Step 3: Set λ = np and write X ≈ Po(λ).

  第三步：令 λ = np，并写出 X ≈ Po(λ)。

• Step 4: Use the Poisson formula, tables, or a calculator with Poisson cumulative distribution functions to calculate the required probability.

  第四步：使用泊松公式、查表或计算器的泊松累积分布函数来计算所需概率。

Always write the approximation symbol ≈ rather than = when rewriting the distribution, because the result is approximate.

在改写分布时，务必使用近似符号 ≈ 而不是等号 =，因为结果是近似值。

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6. Worked Example 1: Individual Probability | 例题一：单点概率

Example: A factory produces components. The probability that a component is defective is p = 0.02. A random sample of n = 200 components is taken. Find the probability that exactly 5 components are defective.

例题：某工厂生产零件，零件为次品的概率为 p = 0.02。现随机抽取 n = 200 个零件，求恰好有 5 个次品的概率。

Solution: Let X be the number of defective components. Then X ~ B(200, 0.02). Here n = 200 ≥ 50, p = 0.02 ≤ 0.1, and λ = 200 × 0.02 = 4 ≤ 5. The conditions are satisfied, so we use X ≈ Po(4).

解答：设 X 为次品数，则 X ~ B(200, 0.02)。这里 n = 200 ≥ 50，p = 0.02 ≤ 0.1，且 λ = 200 × 0.02 = 4 ≤ 5，满足条件，因此用 X ≈ Po(4)。

Using the Poisson probability formula:

P(X = 5) = e⁻⁴4⁵ / 5! = e⁻⁴ × 1024 / 120 ≈ 0.1563

So the approximate probability of exactly 5 defective components is about 0.1563.

使用泊松概率公式：

P(X = 5) = e⁻⁴4⁵ / 5! = e⁻⁴ × 1024 / 120 ≈ 0.1563

因此，恰好有 5 个次品的近似概率约为 0.1563。

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7. Worked Example 2: Cumulative Probability | 例题二：累积概率

Example: In a large population, the probability that a person has a rare blood type is 0.008. In a random sample of 500 people, find the probability that at most 3 people have this blood type.

例题：某大型人群中，一个人拥有稀有血型的概率为 0.008。现随机抽取 500 人，求最多有 3 人拥有该血型的概率。

Solution: Let X be the number of people with the rare blood type. Then X ~ B(500, 0.008). Here n = 500 ≥ 50, p = 0.008 ≤ 0.1, and λ = 500 × 0.008 = 4. The conditions are met, so X ≈ Po(4).

解答：设 X 为拥有稀有血型的人数，则 X ~ B(500, 0.008)。这里 n = 500 ≥ 50，p = 0.008 ≤ 0.1，且 λ = 500 × 0.008 = 4，满足条件，因此 X ≈ Po(4)。

We need P(X ≤ 3). Adding the first four Poisson probabilities:

P(X ≤ 3) = e⁻⁴(4⁰/0! + 4¹/1! + 4²/2! + 4³/3!)

P(X ≤ 3) = e⁻⁴(1 + 4 + 8 + 10.6667) ≈ 0.4335

Thus the probability that at most 3 people have the rare blood type is approximately 0.4335.

我们需要 P(X ≤ 3)。将泊松分布前四项概率相加：

P(X ≤ 3) = e⁻⁴(4⁰/0! + 4¹/1! + 4²/2! + 4³/3!)

P(X ≤ 3) = e⁻⁴(1 + 4 + 8 + 10.6667) ≈ 0.4335

因此，最多有 3 人拥有稀有血型的概率约为 0.4335。

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8. Accuracy and Comparison with Normal Approximation | 精确度与正态近似的比较

The Poisson approximation is excellent when n is large and p is small. For example, B(100, 0.05) compared with Po(5) gives probabilities that agree to several decimal places. As n increases and p decreases, the approximation becomes more accurate.

当 n 很大且 p 很小时，泊松近似效果极佳。例如，B(100, 0.05) 与 Po(5) 的概率结果可精确到多位小数。随着 n 增大、p 减小，近似越来越准确。

However, if λ = np is large, usually greater than 5 or 10, the Poisson distribution becomes more symmetric and the normal approximation with continuity correction is often preferred. The normal approximation is also appropriate when both np ≥ 5 and n(1 − p) ≥ 5.

然而，如果 λ = np 较大，通常大于 5 或 10，泊松分布会变得对称，此时使用带连续性修正的正态近似往往更好。当 np ≥ 5 且 n(1 − p) ≥ 5 时，正态近似也适用。

One key difference: when approximating a binomial distribution by a normal distribution, a continuity correction is required because the normal distribution is continuous and the binomial is discrete. The Poisson distribution is also discrete, so no continuity correction is needed when using the Poisson approximation.

一个关键区别：用正态分布近似二项分布时，由于正态是连续分布而二项是离散分布，因此需要进行连续性修正；而泊松分布同样是离散分布，所以使用泊松近似时不需要连续性修正。

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9. Common Pitfalls and Exam Tips | 常见错误与考试提示

Students often lose marks in this topic by overlooking the conditions or substituting incorrectly. The following points are especially important:

同学们在这一主题中常因忽略条件或代错参数而失分。以下几点尤其重要：

• Always check n ≥ 50 and p ≤ 0.1 before applying the approximation. If the question does not state these values, you should still verify them.

  在应用近似前，务必检查 n ≥ 50 和 p ≤ 0.1。即使题目没有明确给出，你也应当先验证。

• Set λ = np, not λ = p and not λ = n. This is a frequent source of error.

  令 λ = np，而不是 λ = p 或 λ = n。这是常见的错误来源。

• If λ is much larger than 5, the Poisson approximation may be unreliable; consider whether a normal approximation is expected.

  如果 λ 远大于 5，泊松近似可能不可靠；此时需要考虑是否应使用正态近似。

• Use the approximation symbol ≈, not =, when replacing B(n, p) with Po(λ).

  用 Po(λ) 替换 B(n, p) 时，使用近似符号 ≈，不要写成 =。

• On a calculator, use the Poisson cumulative distribution function to find values such as P(X ≤ x) quickly, and remember the complement rule P(X ≥ x) = 1 − P(X ≤ x − 1).

  在计算器上，使用泊松累积分布函数可以快速求出 P(X ≤ x) 等值，并记住补事件法则 P(X ≥ x) = 1 − P(X ≤ x − 1)。

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10. Summary | 总结

The Poisson approximation to the binomial distribution is a valuable tool when n is large and p is small. The table below summarises the key concepts:

泊松分布近似二项分布是当 n 大、p 小时非常实用的工具。下表总结了关键要点：

Aspect	Binomial B(n, p)	Poisson Approximation Po(λ)
Mean	np	λ = np
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