Using the Poisson Distribution to Approximate the Binomial Distribution | 使用泊松分布近似二项分布

📚 Using the Poisson Distribution to Approximate the Binomial Distribution | 使用泊松分布近似二项分布

In A-Level statistics, the binomial distribution can involve very large powers and binomial coefficients that are difficult to evaluate. When n is large and p is small, the Poisson distribution provides a much simpler approximation because it depends only on the single parameter λ = np.

在 A-Level 统计中，二项分布可能涉及非常大的幂和组合数，计算困难。当 n 很大且 p 很小时，泊松分布能提供一种简单得多的近似，因为它只依赖单一参数 λ = np。

1. When the Poisson Approximation Is Valid | 泊松近似何时适用

Suppose X ~ B(n, p). The Poisson approximation replaces this with X ≈ Po(λ), where λ = np. It is used when n is large, p is small, and the product np is moderate, so that the binomial probabilities become very close to Poisson probabilities.

假设 X ~ B(n, p)。泊松近似将 X 近似为 Po(λ)，其中 λ = np。当 n 很大、p 很小且乘积 np 适中时，可以使用此近似，此时二项概率会非常接近泊松概率。

• n is large (commonly n ≥ 50) | n 很大（通常 n ≥ 50）
• p is small (commonly p ≤ 0.1) | p 很小（通常 p ≤ 0.1）
• λ = np is moderate (often λ < 10) | λ = np 适中（通常 λ < 10）

2. The Single Parameter λ = np | 单一参数 λ = np

For the binomial distribution, E(X) = np and Var(X) = np(1 − p). When p is small, 1 − p ≈ 1, so Var(X) ≈ np = λ. The Poisson distribution Po(λ) has both mean and variance equal to λ, which is why it can match the binomial very closely under these conditions.

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

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