A-Level Mathematics: Using the Normal Distribution to Approximate the Binomial Distribution | A-Level数学：用正态分布近似二项分布

📚 A-Level Mathematics: Using the Normal Distribution to Approximate the Binomial Distribution | A-Level数学：用正态分布近似二项分布

The binomial distribution is one of the most important discrete distributions in A-Level statistics. It gives the probability of exactly \(k\) successes in \(n\) independent trials, each with success probability \(p\). However, as \(n\) becomes large, exact binomial calculations can involve huge coefficients and long sums. The normal distribution, which is continuous and symmetric, can provide a highly accurate approximation when certain conditions are met.

二项分布是A-Level统计学中最重要的离散分布之一。它描述在\(n\)次独立试验中恰好出现\(k\)次成功的概率，其中每次试验的成功概率为\(p\)。随着\(n\)增大，精确二项计算往往涉及巨大的组合数和冗长的求和。在满足一定条件时，连续且对称的正态分布可以非常准确地近似二项概率。

1. Why Do We Need a Normal Approximation? | 为什么需要正态近似？

When \(n\) is large, calculating binomial probabilities such as \(P(20 \le X \le 40)\) can require many terms. The binomial coefficient \(\binom{n}{k}\) grows rapidly, and summing dozens of terms by hand is tedious and often not required in examinations.

当\(n\)很大时，像\(P(20 \le X \le 40)\)这样的二项概率需要计算很多项。组合数\(\binom{n}{k}\)增长很快，手工求和十分麻烦，考试中通常也不需要逐项计算。

Instead, we use the fact that the binomial distribution is the sum of independent Bernoulli trials. By the Central Limit Theorem, the sum of many independent random variables is approximately normal. This means we can replace the discrete binomial distribution with a continuous normal curve that has the same mean and variance.

因此，我们利用一个重要事实：二项分布是若干独立伯努利试验之和。根据中心极限定理，许多独立随机变量之和近似服从正态分布。这意味着我们可以用一条具有相同均值和方差的正态曲线来代替离散的二项分布。

2. The Approximation Rule and Its Conditions | 近似规则及适用条件

If \(X \sim B(n,p)\), then the mean and variance of \(X\) are:

\(\mu = E(X) = np\)，\(\sigma^2 = Var(X) = np(1-p)\)

The normal approximation is therefore written as:

\(X \approx N(np, np(1-p))\)

This approximation should only be used when the binomial distribution is reasonably symmetric and not too skewed. The standard check for A-Level is:

• \(np > 5\) and \(n(1-p) > 5\). Some textbooks use \(np \ge 5\) and \(n(1-p) \ge 5\).

  A-Level标准检查条件是：\(np > 5\)且\(n(1-p) > 5\)。有些教材使用\(np \ge 5\)且\(n(1-p) \ge 5\)。

• The closer \(p\) is to 0.5, the more symmetric the binomial distribution becomes, and the better the normal approximation works.

  \(p\)越接近0.5，二项分布越对称，正态近似效果越好。

• If \(p\) is very close to 0 or 1, the binomial distribution is highly skewed, and the normal approximation may be unreliable unless \(n\) is extremely large.

  如果\(p\)非常接近0或1，二项分布会严重偏斜，除非\(n\)极大，否则正态近似可能并不可靠。

3. Why Do We Need the Continuity Correction? | 为什么要使用连续性修正？

The binomial distribution is discrete: \(X\) can only take integer values such as 0, 1, 2, … . The normal distribution is continuous, so the area at a single point is exactly zero. If we want to approximate \(P(X = k)\), we cannot simply read the height of the normal curve. Instead, we assign the probability of \(k\) to the interval from \(k – 0.5\) to \(k + 0.5\). This adjustment of 0.5 is called the continuity correction.

二项分布是离散的：\(X\)只能取0、1、2等整数值。而正态分布是连续的，单点处的面积恰好为0。如果我们想近似\(P(X = k)\)，不能直接看正态曲线的高度。相反，我们把\(k\)这一点的概率分配到区间\(k – 0.5\)到\(k + 0.5\)上。这0.5的调整量称为连续性修正。

The key idea is:

\(P(X = k) \approx P(k – 0.5 \le Y \le k + 0.5)\)，其中 \(Y \sim N(np, np(1-p))\)

Every time an endpoint of the binomial probability is an integer, we must move it by 0.5 in the direction that makes the interval wider.

每次二项概率的端点取整数时，我们都要朝使区间变宽的方向移动0.5。

4. The Continuity Correction Table | 连续性修正对照表

The table below shows how to convert binomial probability statements into normal probability statements. Here \(a\) and \(b\) are integers, and \(Y\) is the normal approximation

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