📚 Mean and Variance of the Binomial Distribution | 二项分布的均值与方差
In A-Level Statistics, the binomial distribution is one of the most important discrete probability models. This article explains how to find and interpret its mean and variance, including the proofs and exam-style applications.
在 A-Level 统计中，二项分布是最重要的离散概率模型之一。本文讲解如何求并解释其均值与方差，包括证明和考试风格的应用。
1. The Binomial Setting | 二项分布的条件
A binomial distribution arises when we repeat a fixed number of independent trials, each trial has exactly two outcomes called ‘success’ and ‘failure’, and the probability of success p remains constant from trial to trial.
当我们重复固定次数的独立试验，每次试验只有两个结果，称为“成功”和“失败”，且每次试验的成功概率 p 保持不变时，就得到二项分布。
We write X ~ B(n, p), where n is the number of trials and p is the probability of success in one trial. The number of failures is often denoted by q = 1 − p.
我们记作 X ~ B(n, p)，其中 n 是试验次数，p 是单次试验成功的概率。失败的概率通常用 q = 1 − p 表示。
2. Why Mean and Variance Matter | 为什么均值和方差重要
The mean of a random variable tells us the average value we expect over many repetitions, while the variance measures the spread or variability around that mean.
随机变量的均值告诉我们在多次重复中预期的平均值，而方差衡量围绕该均值的离散程度或变异性。
For the binomial distribution, both the mean and variance depend only on n and p. This means we can describe the centre and spread without listing the entire probability distribution.
对于二项分布，均值和方差仅由 n 和 p 决定。这意味着我们无需列出整个概率分布，就可以描述其中心和离散程度。
3. Expected Value of a Binomial Random Variable | 二项随机变量的期望值
For a binomial random variable X ~ B(n, p), the expected value, or mean, is given by the product of the number of trials and the probability of success.
对于二项随机变量 X ~ B(n, p)，期望值（即均值）由试验次数与成功概率的乘积给出。
E(X) = np
This formula makes sense: if you carry out n independent attempts, each with success probability p, the average number of successes should be n times p.
这个公式很直观：如果进行 n 次独立尝试，每次成功概率为 p，那么平均成功次数应该是 n 乘以 p。
4. Variance and Standard Deviation of a Binomial Random Variable | 二项随机变量的方差与标准差
The variance of a binomial random variable is np multiplied by 1 − p, and the standard deviation is the square root of the variance.
二项随机变量的方差为 np 乘以 1 − p，标准差是方差的平方根。
Var(X) = np(1 − p) = npq
σ = √(np(1 − p))
Here σ is the standard deviation. Notice that the variance is largest when p = 0.5, because the outcome is most uncertain when success and failure are equally likely.
这里 σ 是标准差。注意当 p = 0.5 时方差最大，因为成功与失败等可能时结果最不确定。
5. Proof of the Mean Using Linearity of Expectation | 用期望线性性质证明均值
A useful proof defines an indicator variable Iᵢ for each trial, where Iᵢ = 1 if trial i is a success and Iᵢ = 0 otherwise.
一个常用的证明方法是为每次试验定义指示变量 Iᵢ，若第 i 次试验成功，则 Iᵢ = 1，否则 Iᵢ = 0。
For each indicator variable, E(Iᵢ) = 0 × P(Iᵢ = 0) + 1 × P(Iᵢ = 1) = p. Since X = I₁ + I₂ + … + Iₙ, the linearity of expectation gives E(X) = E(I₁) + E(I₂) + … + E(Iₙ) = np.
对于每个指示变量，E(Iᵢ) = 0 × P(Iᵢ = 0) + 1 × P(Iᵢ = 1) = p。因为 X = I₁ + I₂ + … + Iₙ，由期望的线性性质可得 E(X) = E(I₁) + E(I₂) + … + E(Iₙ) = np。
6. Proof of the Variance Using Independence | 用独立性证明方差
We can also use the same indicator variables to prove the variance formula. For one trial, the variance of Iᵢ is p(1 − p) because Iᵢ takes only the values 0 and 1.
我们还可以用相同的指示变量证明方差公式。对于单次试验，Iᵢ 的方差为 p(1 − p)，因为 Iᵢ 只取 0 和 1 两个值。
Since the trials are independent, the variances add: Var(X) = Var(I₁) + Var(I₂) + … + Var(Iₙ). Therefore, Var(X) = np(1 − p).
由于各次试验相互独立，方差可以相加：Var(X) = Var(I₁) + Var(I₂) + … + Var(Iₙ)。因此，Var(X) = np(1 − p)。
This independence step is essential; if the trials were not independent, we could not simply add the variances.
独立这一步非常关键；如果试验不独立，就不能简单地相加方差。
7. Worked Example 1: Calculating Mean and Variance | 示例 1：计算均值与方差
A fair die is rolled 12 times. A success is defined as rolling a six, so n = 12 and p = 1/6. Find the mean, variance and standard deviation of the number of sixes.
一个公平的骰子被掷 12 次。掷出 6 定义为成功，因此 n = 12，p = 1/6。求掷出 6 的次数的均值、方差和标准差。
The mean is E(X) = np = 12 × 1/6 = 2, so on average we expect two sixes in 12 rolls.
均值为 E(X) = np = 12 × 1/6 = 2，因此平均而言，12 次投掷中预期出现两个 6。
The variance is Var(X) = np(1 − p) = 12 × 1/6 × 5/6 = 60/36 = 5/3 ≈ 1.67. The standard deviation is √(5/3) ≈ 1.29.
方差为 Var(X) = np(1 − p) = 12 × 1/6 × 5/6 = 60/36 = 5/3 ≈ 1.67。标准差为 √(5/3) ≈ 1.29。
8. Worked Example 2: Using Mean and Variance to Find n and p | 示例 2：用均值与方差求 n 和 p
Suppose X ~ B(n, p) has E(X) = 8 and Var(X) = 4.8. Find the values of n and p.
假设 X ~ B(n, p)，已知 E(X) = 8，Var(X) = 4.8。求 n 和 p 的值。
From the mean formula, np = 8. From the variance formula, np(1 − p) = 4.8. Substituting np = 8 gives 8(1 − p) = 4.8, so 1 − p = 0.6 and p = 0.4.
由均值公式得 np = 8。由方差公式得 np(1 − p) = 4.8。将 np = 8 代入可得 8(1 − p) = 4.8，因此 1 − p = 0.6，p = 0.4。
Then n = 8 / p = 8 / 0.4 = 20. So the distribution is X ~ B(20, 0.4).
然后 n = 8 / p = 8 / 0.4 = 20。因此该分布为 X ~ B(20, 0.4)。
9. Interpreting the Standard Deviation and Spread | 解释标准差与离散程度
The standard deviation √(np(1 − p)) measures how far individual outcomes typically fall from the mean. A larger n gives a larger absolute spread, but the relative spread compared with the mean becomes smaller.
标准差 √(np(1 − p)) 衡量单个结果通常距离均值有多远。较大的 n 会带来更大的绝对离散度，但与均值相比的相对离散度会变小。
For example, if n = 100 and p = 0.5, then E(X) = 50 and σ = √(100 × 0.5 × 0.5) = 5. Most observations fall within roughly two standard deviations of the mean, so between 40 and 60.
例如，若 n = 100 且 p = 0.5，则 E(X) = 50，σ = √(100 × 0.5 × 0.5) = 5。大多数观测值大约落在均值的两个标准差范围内，即 40 到 60 之间。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
Students often forget that the variance uses p(1 − p) rather than p alone. Writing Var(X) = np is a common error, but this would not include the uncertainty from failures.
学生常常忘记方差使用的是 p(1 − p) 而不是单独的 p。把 Var(X) = np 写出来是一个常见错误，但这样做没有包含失败带来的不确定性。
Another mistake is adding variances when trials are not independent. Only independent binomial trials allow the simple addition Var(X) = np(1 − p).
另一个错误是在试验不独立时仍然相加方差。只有独立的二项试验才能使用简单的加法 Var(X) = np(1 − p)。
Exam tip: check that your variance is smaller than your mean, since np(1 − p) < np when 0 < p < 1. If the variance is larger than the mean, your values of n and p may be wrong.
应试技巧：检查方差是否小于均值，因为当 0 < p < 1 时，np(1 − p) < np。如果方差大于均值，你的 n 和 p 值可能有误。
11. Summary and Key Formulae | 小结与关键公式
For a binomial random variable X ~ B(n, p), the key results are the mean and variance. These allow quick calculation of the centre and spread of the distribution.
对于二项随机变量 X ~ B(n, p)，关键结果是均值和方差。它们能快速计算分布的中心和离散程度。
E(X) = np
Var(X) = np(1 − p) = npq
SD(X) = √(np(1 − p))
Remember that these formulae are valid only when the trials are independent and p is constant. Use them confidently in Edexcel exam questions involving binomial distributions.
请记住，这些公式仅在试验相互独立且 p 保持不变时有效。在涉及二项分布的 Edexcel 考试题中，可以放心使用这些公式。
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