📚 Statistical Distributions | 统计分布
Statistical distributions describe how probabilities are spread across the possible values of a random variable. In the Edexcel A-Level Statistics specification you must recognise discrete distributions such as the binomial and the Poisson, and continuous distributions such as the normal. You must also be able to calculate probabilities, expectations and variances, and to apply approximations correctly when the exact distribution is awkward to use.
统计分布描述随机变量在其所有可能取值上的概率分布情况。在爱德思 A-Level 统计大纲中，你需要掌握离散分布（如二项分布和泊松分布）以及连续分布（如正态分布），并能正确计算概率、期望和方差，在精确分布难以使用时要能合理选用近似方法。
1. Random Variables and Probability Distributions | 随机变量与概率分布
A random variable X is a numerical quantity whose value is determined by the outcome of a random experiment - for example, the number of heads obtained when three coins are tossed. If the possible values can be listed one by one, such as 0, 1, 2, 3, then X is a discrete random variable. If X can take any value in an interval, for example the time a student spends on a question, then X is a continuous random variable.
随机变量 X 表示由随机试验结果决定的数值，例如抛三枚硬币时正面出现的次数。如果它的取值可以一一列出，如 0、1、2、3，则称其为离散随机变量；如果它可以在某一区间内取任意值，例如学生做一道题所用的时间，则称其为连续随机变量。
The probability distribution of X gives the probability attached to each possible value. For a discrete distribution there are two essential conditions: every probability must lie between 0 and 1 inclusive, and the total probability over all possible values must equal 1.
X 的概率分布给出每一个可能取值所对应的概率。对离散分布而言有两个必要条件：每个概率都必须介于 0 与 1 之间（含端点），且所有可能取值的概率之和必须等于 1。
0 ≤ P(X = x) ≤ 1 and Σ P(X = x) = 1
ties of all values up to and including x. It is the quickest route to questions phrased as 'at most', 'at least' or 'no more than', and it is also the bridge to the complement rule P(X ≥ r) = 1 − P(X ≤ r − 1).
累积分布函数 F(x) = P(X ≤ x) 表示所有不超过 x 的取值的概率之和。它是解决"至多""至少""不多于"这类问题的最快途径，同时也是通向补事件法则 P(X ≥ r) = 1 − P(X ≤ r − 1) 的桥梁。
2. Discrete Distributions and Probability Functions | 离散分布与概率函数
sually given as a table, as a formula such as P(X = r) = k r, or in words, and you are then asked to find the unknown constant k by forcing the probabilities to add to 1.
离散分布由其概率函数 P(X = r) 对每一个可能取值 r 完全确定。在考题中，这个函数通常以表格形式、以 P(X = r) = k r 这样的公式形式或文字形式给出，随后要求你利用概率之和为 1 这一条件求出未知常数 k。
dual probability, any cumulative probability and any expectation can be obtained directly.
解题模板：若对 r = 1, 2, 3, 4 有 P(X = r) = k r，则 k(1 + 2 + 3 + 4) = 10k = 1，故 k = 0.1。由此可直接求出任一概率、任一累积概率以及期望。
In that case the sum Σ P(X = r) becomes an infinite series whose total must still equal 1.
要注意，离散随机变量也可能取无穷多个值，例如直到首次成功所需的试验次数。此时求和 Σ P(X = r) 变成无穷级数，但其总和仍必须等于 1。
3. Expectation and Variance | 期望与方差
The expectation E(X) is the long-run average value of X, and the variance Var(X) measures how spread out the values are around that average. For a discrete random variable both are defined by weighted sums, where each value x is weighted by its probability.
期望 E(X) 是 X 的长期平均值，方差 Var(X) 衡量取值围绕该平均值的分散程度。对离散随机变量而言，两者都由加权和定义，其中每个取值 x 以其概率为权重。
E(X) = Σ x P(X = x) Var(X) = Σ x² P(X = x) − (E(X))²
efore. A common slip is to compute Σ x² P(X = x) and forget the −(E(X))² term altogether, which produces a variance that is too large.
第二个公式几乎总比 Var(X) = E((X − μ)²) 更快，而且一定要先算期望的平方再相减，顺序不能颠倒。常见错误是只算出 Σ x² P(X = x) 就忘记减去 (E(X))²，导致方差偏大。
the distribution without altering its spread, while the multiplier a scales the spread by |a|. Standard deviation is always the positive square root of the variance, σ = √Var(X).
线性变换有简洁的规律：E(aX + b) = a E(X) + b，但 Var(aX + b) = a² Var(X)。常数 b 只平移分布而不改变离散程度，而系数 a 使离散程度按 |a| 缩放。标准差始终是方差的正平方根，σ = √Var(X)。
4. The Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials when each trial has exactly two outcomes and the probability of success stays constant. If X counts successes in n trials with success probability p, we write X ~ B(n, p).
二项分布描述在固定次数的独立试验中成功的次数，其中每次试验只有两种结果，且成功概率保持不变。若 X 表示 n 次试验中的成功次数、单次成功概率为 p，则记作 X ~ B(n, p)。
The four conditions to quote are: a fixed number of trials n; each trial independent of the others; only two possible outcomes per trial (success or failure); and a constant probability p of success on every trial. If any one of these fails - for instance sampling without replacement from a small population - the binomial model is not valid.
需要写出的四个条件是：试验次数 n 固定；各次试验相互独立；每次试验只有两种结果（成功或失败）；每次试验成功的概率 p 恒定。若其中任一条件不满足，例如从很小的总体中不放回抽样，则二项模型不成立。
P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ E(X) = np Var(X) = np(1 − p)
The factor C(n, r) counts the number of orders in which the r successes can appear, pʳ gives the probability of those successes and (1 − p)ⁿ⁻ʳ cumulative binomial tables or your calculator rather than adding individual terms by hand.
因子 C(n, r) 统计 r 次成功可能出现的排列方式数，pʳ 给出这些成功的概率，(1 − p)ⁿ⁻ʳ 给出其余失败的概率。对于 P(X ≤ 3) 或 P(X ≥ 5) 这类累积问题，应使用累积二项分布表或计算器，而不是逐项手算相加。
5. The Poisson Distribution | 泊松分布
The Poisson distribution models the number of occurrences of a rare event in a fixed interval of time, length, area or volume, when occurrences happen independently and at a constant average rate. If the mean number of occurrences in the interval is λ, we write X ~ Po(λ).
泊松分布描述稀有事件在固定的时间、长度、面积或体积区间内发生的次数，其前提是事件独立发生且平均发生率恒定。若该区间内事件发生次数的平均值为 λ，则记作 X ~ Po(λ)。
Typical contexts include the number of typing errors on a page, the number of calls arriving at a switchboard in one minute, or the number of defective items in a large batch. The key modelling assumption is that the probability of an occurrence in a very short interval is proportional to the length of that interval, and that two occurrences cannot happen at exactly the same instant.
典型情境包括：一页纸上的打字错误数、一分钟内到达总机的电话数、一大批产品中的次品数。其核心建模假设是：在极短区间内发生一次事件的概率与该区间长度成正比，且两次事件不会在同一瞬间发生。
P(X = r) = e⁻λ × λʳ ÷ r! E(X) = Var(X) = λ
A distinctive feature of the Poisson distribution is that its mean and variance are equal, both being λ. A useful consequence is that if occurrences in two disjoint intervals are independent, then the total count follows a Poisson distribution whose parameter is the sum of the individual parameters.
泊松分布的一个显著特点是其均值与方差相等，都等于 λ。一个有用的推论是：若两个互不重叠区间内的事件发生相互独立，则总次数服从参数为两个参数之和的泊松分布。
6. The Normal Distribution | 正态分布
The normal distribution is the standard continuous model: a symmetric bell-shaped curve that is fully determined by its mean μ and its variance σ². We write X ~ N(μ, σ²), and the total area under the curve equals 1, so probabilities correspond to areas.
正态分布是标准的连续型模型：一条对称的钟形曲线，完全由均值 μ 和方差 σ² 确定。记作 X ~ N(μ, σ²)，曲线下总面积为 1，因此概率对应于面积。
ymmetric about μ, we always have P(X < μ − d) = P(X > μ + d), and P(X > μ) = P(X < μ) = 0.5.
对连续随机变量而言，P(X 等于某个单点值) = 0，因此 P(X < a) 与 P(X ≤ a) 相等，端点取不取都无所谓。由于分布关于 μ 对称，总有 P(X < μ − d) = P(X > μ + d)，且 P(X > μ) = P(X < μ) = 0.5。
The empirical rule gives quick benchmarks: about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. So roughly 95% of values fall in the interval from μ − 2σ to μ + 2σ, which is often enough to judge whether an observation is unusual.
经验法则给出快速参照：约 68% 的取值落在均值一个标准差范围内，约 95% 落在两个标准差内，约 99.7% 落在三个标准差内。也就是说，约 95% 的取值位于 μ − 2σ 到 μ + 2σ 之间，这常足以判断某个观测值是否异常。
7. Standardising and Using z-Values | 标准化与 z 值的使用
Because normal tables are printed only for the standard normal distribution Z ~ N(0, 1), every problem must first be converted by standardisation. The z-value tells you how many standard deviations a value lies above or below the mean.
由于正态分布表只给出标准正态分布 Z ~ N(0, 1) 的数值，任何问题都必须先通过标准化转换。z 值表示某个取值高于或低于均值多少个标准差。
z = (x − μ) ÷ σ equivalently x = μ + zσ
Sketch the curve before you calculate, shading the region you want. If the shaded area is on the left of a value, read the table directly; if it is on the right, subtract from 1; if it lies between two values, subtract the smaller cumulative probability from the larger.
计算前先画出曲线草图，并把你要求的区域涂上阴影。若阴影在某值左侧，直接查表；若在右侧，用 1 相减；若夹在两个值之间，则用较大的累积概率减去较小的累积概率。
l distribution and then rearrange x = μ + zσ. Remember that if x is below the mean, z is negative, and the table entry should be read using the symmetry of the standard normal curve.
有些题目反向求解：已知概率，要求出 x。此时应从正态分布百分位表中查出相应的 z 值，再通过 x = μ + zσ 反解。要记住，若 x 低于均值则 z 为负，查表时应利用标准正态曲线的对称性。
8. The Normal Approximation to the Binomial | 二项分布的正态近似
mated by a normal distribution with the same mean and variance: μ = np and σ² = np(1 − p).
当 n 很大且 p 接近 0.5 时，二项分布表难以查用，但可用具有相同均值和方差的正态分布很好地近似：μ = np，σ² = np(1 − p)。
Use the approximation only when both np and n(1 − p) exceed about 5, so that the distribution is not strongly skewed. State these checks explicitly in your answer, as they usually earn a method mark.
只有当 np 与 n(1 − p) 都大于约 5 时才使用该近似，以保证分布没有明显偏斜。答题时要明确写出这些检验条件，因为它们通常对应一个方法分。
). Strict inequalities shift the other way, so P(X < r) becomes P(Y ≤ r − 0.5). Forgetting the half-unit is one of the most frequent errors in this topic.
由于你是用光滑曲线替代宽度为 1 的条形，需要做连续性修正：P(X ≤ r) 变为 P(Y ≤ r + 0.5)，P(X ≥ r) 变为 P(Y ≥ r − 0.5)。严格不等号则方向相反，例如 P(X < r) 变为 P(Y ≤ r − 0.5)。忘记这半个单位的修正，是本主题最常见的错误之一。
9. The Poisson Approximation to the Binomial | 二项分布的泊松近似
stribution is well approximated by the Poisson distribution Po(λ). This is the mathematical reason why the Poisson distribution models rare events.
当 n 很大、p 很小而乘积 λ = np 适中时，二项分布可以用泊松分布 Po(λ) 很好地近似。这正是泊松分布能刻画稀有事件的数学原因。
hecking that the average number of successes per batch stays small. State the values of n and p you are using and the resulting λ before you calculate.
常用的经验判据是：当 n ≥ 50、p ≤ 0.1 且 np < 10 时使用该近似，并确认每批中成功次数的平均值较小。计算前应先说明所用的 n、p 以及由此得到的 λ。
Note carefully that both approximations compete: the normal approximation suits p near 0.5, while the Poisson approximation suits small p and large n. Choosing the right one, and justifying the choice, is frequently tested in exam questions on modelling assumptions.
要特别注意两种近似各有适用范围：正态近似适合 p 接近 0.5 的情形，泊松近似适合 p 很小而 n 很大的情形。选择正确的方法并说明理由，是考试中关于建模假设的常考点。
10. Sums and Linear Combinations | 分布的和与线性组合
If X and Y are independent and both normally distributed, then X + Y is also normally distributed, with the means added and the variances added. Critically, you add variances, not standard deviations, so you must square the standard deviations first.
若 X 与 Y 相互独立且都服从正态分布，则 X + Y 也服从正态分布：均值相加，方差相加。关键是要加方差而不是标准差，因此必须先把标准差平方。
X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²) aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)
ason a confidence interval for a difference of two means uses the sum, not the difference, of the variances.
对差值 X − Y，均值相减但方差仍然相加，因为 Var(−Y) = Var(Y)。这也是两个均值之差的置信区间要用方差之和、而不是方差之差的原因。
Similar additivity holds for discrete models: if X ~ B(n, p) and Y ~ B(m, p) are independent, then X + Y ~ B(n + m, p); and if X ~ Po(λ) and Y ~ Po(μ) are independent, then X + Y ~ Po(λ + μ). Both results require the same underlying parameter, which is exactly the condition most students overlook.
离散模型也有类似的相加性：若 X ~ B(n, p) 与 Y ~ B(m, p) 独立，则 X + Y ~ B(n + m, p)；若 X ~ Po(λ) 与 Y ~ Po(μ) 独立，则 X + Y ~ Po(λ + μ)。两个结论都要求参数相同，而这恰恰是多数学生容易忽略的条件。
11. Exam Technique and Common Errors | 考试技巧与常见错误
Write down the distribution you are using, with its parameters, before doing any arithmetic: a line such as X ~ B(20, 0.3), X ~ Po(4.5) or X ~ N(62, 12²) secures the modelling mark even if a later calculation slips. Then decide whether the question needs an exact distribution or an approximation, and justify the choice.
在动手计算之前，先写出所用的分布及其参数：像 X ~ B(20, 0.3)、X ~ Po(4.5) 或 X ~ N(62, 12²) 这样的一行，即使后续计算出错也能拿到建模分。然后再判断题目需要精确分布还是近似，并说明理由。
Recurring error list: mixing up variance and standard deviation in the normal distribution; omitting the continuity correction; forgetting the −(E(X))² term in the variance of a discrete distribution; using the binomial when sampling is without replacement from a small population; and using a Poisson when the mean rate is not constant over the interval.
常见错误清单：在正态分布中混淆方差与标准差；遗漏连续性修正；计算离散分布方差时忘记减去 (E(X))²；在小总体不放回抽样时误用二项分布；以及在区间内平均发生率不恒定时误用泊松分布。
Finally, interpret your answer in context. A probability alone is worth little; saying that there is about a 3% chance of observing at least this many defective items, and that this is therefore unusual, turns a calculation into a conclusion.
最后要结合情境解释结果。单纯给出一个概率意义有限；若说明"观测到至少这么多次品的概率约为 3%，因此属于异常情况"，就把计算变成了结论。
12. Summary of Key Formulae | 关键公式总结
The table below collects the results that appear most often in Edexcel A-Level Statistics. Learn the conditions as well as the formulae, because examiners reward correct modelling assumptions explicitly.
下表汇总了爱德思 A-Level 统计中最常出现的结果。既要记住公式，也要记住适用条件，因为考官会为正确的建模假设单独给分。
| Model | 模型 | Notation | 记号 | Probability | 概率 | Mean, Variance | 均值、方差 |
|---|---|---|---|
| Binomial | 二项分布 | B(n, p) | C(n, r) pʳ (1 − p)ⁿ⁻ʳ | np, np(1 − p) |
| Poisson | 泊松分布 | Po(λ) | e⁻λ λʳ ÷ r! | λ, λ |
| Normal | 正态分布 | N(μ, σ²) | style="border:1px
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