📚 Probability Distributions and Tables | 概率分布与统计表
In IB Mathematics, the study of probability distributions and the use of statistical tables is essential for modelling random phenomena and making predictions. A probability distribution describes how the probabilities are distributed over the possible values of a random variable. Tables provide a convenient way to present these probabilities, and for well‑known families such as the binomial and normal distributions, printed tables allow us to find probabilities without lengthy calculations. This article covers discrete random variables, expected value, variance, the binomial distribution, the normal distribution, and how to read and apply the standard statistical tables that appear in the IB formula booklet and examinations.
在 IB 数学中，概率分布及统计用表的学习对于建立随机现象模型和进行预测至关重要。概率分布描述了概率如何分布在随机变量的各个可能取值上。表格是一种便捷的表达方式；对于二项分布、正态分布等常见分布族，统计表能让我们在不进行繁琐计算的情况下直接找到概率。本文将涵盖离散随机变量、期望值、方差、二项分布、正态分布，以及如何阅读和使用 IB 公式手册及考试中出现的标准统计表。
1. Discrete Random Variables and Probability Distribution Tables | 离散随机变量与概率分布表
A discrete random variable X takes a countable number of possible values x₁, x₂, … . Its probability distribution lists each possible value together with its probability P(X = x). The distribution can be displayed compactly in a probability distribution table, where one row (or column) gives the values and the other gives the corresponding probabilities. Two rules must always hold: 0 ≤ P(X = x) ≤ 1 for every outcome, and the sum of all probabilities equals 1.
离散随机变量 X 取一列可数的可能值 x₁, x₂, … 。它的概率分布列出了每个可能取值及其概率 P(X = x)。分布可以用概率分布表紧凑地表示，其中一行（或一列）给出所有取值，另一行给出对应的概率。两个规则永远成立：对每一个结果都有 0 ≤ P(X = x) ≤ 1，并且所有概率之和为 1。
Example – Probability distribution table for the number of heads when tossing three fair coins:
示例 – 掷三枚均匀硬币时正面朝上次数的概率分布表：
| x (Number of heads) | P(X = x) |
|---|---|
| 0 | 1/8 |
| 1 | 3/8 |
| 2 | 3/8 |
| 3 | 1/8 |
From the table we can immediately read off, for example, that the chance of getting exactly two heads is 3/8.
从表中可以直接读出，例如得到恰好两个正面的概率是 3/8。
2. Expected Value E(X) | 期望值 E(X)
The expected value (or mean) of a discrete random variable X, denoted E(X) or μ, is the long‑run average of the variable over many independent repetitions. It is computed as the probability‑weighted sum of the possible values:
E(X) = μ = Σ x·P(X = x)
离散随机变量 X 的期望值（或均值），记作 E(X) 或 μ，是经过大量独立重复试验后该变量的长期平均值。它的计算方法是将每个可能取值与其概率的乘积求和。
For the three‑coin example: E(X) = 0×(1/8) + 1×(3/8) + 2×(3/8) + 3×(1/8) = 1.5. This tells us that on average we expect 1.5 heads per trial.
对于三枚硬币的例子：E(X) = 0×(1/8) + 1×(3/8) + 2×(3/8) + 3×(1/8) = 1.5。这告诉我们每次试验平均会出现 1.5 个正面。
When a probability distribution is given in a table, always check that the probabilities sum to 1 before calculating the expected value. The mean is not necessarily one of the possible values of X; it is a centre of mass of the distribution.
当概率分布以表格给出时，计算期望值前一定要检查概率之和是否为 1。均值不一定等于 X 的某个具体取值，它是分布的质心。
3. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of the distribution – how far the values typically deviate from the mean. It is defined as the expected value of the squared deviations:
Var(X) = Σ (x – μ)²·P(X = x)
方差衡量分布的离散程度——取值通常偏离均值多远。它定义为方差差的平方的期望值。
A computationally convenient form is Var(X) = E(X²) – [E(X)]², where E(X²) = Σ x²·P(X = x). The standard deviation σ is the positive square root of the variance: σ = √Var(X).
一个便于计算的公式为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。标准差 σ 是方差的正平方根：σ = √Var(X)。
For the three‑coin distribution, E(X²) = 0²×1/8 + 1²×3/8 + 2²×3/8 + 3²×1/8 = 3, so Var(X) = 3 – (1.5)² = 0.75, and σ = √0.75 ≈ 0.866. Always remember that variance has squared units while standard deviation has the same units as X.
对于三枚硬币的分布，E(X²) = 0²×1/8 + 1²×3/8 + 2²×3/8 + 3²×1/8 = 3，因此 Var(X) = 3 – (1.5)² = 0.75，σ = √0.75 ≈ 0.866。始终要记得方差的单位是平方单位，而标准差与 X 的单位相同。
4. The Binomial Distribution | 二项分布
A binomial setting arises when there are n independent trials, each having the same probability of success p, and we count the number of successes X. Then X ~ B(n, p). The probability that exactly k successes occur is
P(X = k) = C(n, k) · pᵏ · (1 – p)ⁿ⁻ᵏ
where C(n, k) = n! / [k!·(n – k)!] is the binomial coefficient. The distribution has mean E(X) = n p and variance Var(X) = n p (1 – p).
当存在 n 次独立试验，每次试验成功的概率皆为 p，而我们统计成功次数 X 时，就成为二项分布的情形。此时 X ~ B(n, p)。恰好获得 k 次成功的概率由上述公式给出，其中 C(n, k) 为二项式系数。该分布的期望为 E(X) = n p，方差为 Var(X) = n p (1 – p)。
For example, guessing on a 10‑question multiple‑choice quiz with four options each forms a binomial random variable X ~ B(10, 0.25). The chance of getting exactly 3 correct answers is C(10, 3) · (0.25)³ · (0.75)⁷ ≈ 0.250.
例如，在一份有 10 道四选一选择题的测验中纯靠猜测，得到的答对题数就是一服从二项分布的随机变量 X ~ B(10, 0.25)。恰好答对 3 题的概率为 C(10, 3) · (0.25)³ · (0.75)⁷ ≈ 0.250。
Check the four conditions for a binomial model: fixed number of trials, independence, constant probability, and only two outcomes (success/failure) per trial.
使用二项模型前要检查四个条件：试验次数固定、试验间独立、概率恒定以及每次试验只有两个结果（成功/失败）。
5. Using Binomial Distribution Tables | 使用二项分布表
For small n and common p, IB formula booklets provide cumulative binomial distribution tables, which give P(X ≤ k) rather than individual probabilities. These tables are usually organised with n down the rows and p across the columns; within each block, you look up the value of k along the left side to find the cumulative probability.
对于较小的 n 和常见的 p，IB 公式手册会提供累积二项分布表，表中给出的是 P(X ≤ k) 而非单个概率。这类表格通常以 n 为行、p 为列进行组织；在每个区段内，沿左列找到 k 值即可读出累积概率。
Example – A section of a cumulative binomial table for n = 5 and p = 0.3:
示例 – n = 5, p = 0.3 的累积二项分布表片段：
| k | P(X ≤ k) |
|---|---|
| 0 | 0.1681 |
| 1 | 0.5282 |
| 2 | 0.8369 |
| 3 | 0.9692 |
| 4 | 0.9976 |
| 5 | 1.0000 |
To find P(X = 2) you subtract: P(X ≤ 2) – P(X ≤ 1) = 0.8369 – 0.5282 = 0.3087. To obtain P(X ≥ 2) use the complement: 1 – P(X ≤ 1) = 1 – 0.5282 = 0.4718.
要计算 P(X = 2)，只需做减法：P(X ≤ 2) – P(X ≤ 1) = 0.8369 – 0.5282 = 0.3087。要求 P(X ≥ 2) 则利用补集：1 – P(X ≤ 1) = 1 – 0.5282 = 0.4718。
Always confirm which cumulative probability convention (≤, <, ≥) a given table uses. In IB examinations, tables typically give P(X ≤ k).
始终要确认所用表格的累积概率约定（≤、<或≥）。在 IB 考试中，表格通常提供 P(X ≤ k)。
6. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution that is widely used to model natural variations such as heights, weights, and measurement errors. A normal random variable X ~ N(μ, σ²) has a symmetric bell‑shaped curve centred at the mean μ; the spread is determined by the standard deviation σ. The total area under the curve equals 1, and the probability for an interval is the area between the corresponding x‑values.
正态分布是一种连续型概率分布，广泛用于描述身高、体重和测量误差等自然变异。正态随机变量 X ~ N(μ, σ²) 具有对称的钟形曲线，中心位于均值 μ，散布程度由标准差 σ 决定。曲线下的总面积为 1，一个区间上的概率就是相应 x 值之间的面积。
Properties: about 68% of the data lie within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Any normal distribution can be standardised using Z = (X – μ) / σ, producing Z ~ N(0, 1²).
性质：大约 68% 的数据落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。任何正态分布都可以通过 Z = (X – μ) / σ 标准化，得到 Z ~ N(0, 1²)。
The probability density function is f(x) = 1/(σ√(2π)) · e⁻⁽ˣ⁻µ⁾²/⁽²σ²⁾, but you will rarely use this directly; instead you rely on standard normal tables or your GDC.
概率密度函数为 f(x) = 1/(σ√(2π)) · e⁻⁽ˣ⁻µ⁾²/⁽²σ²⁾，但你很少会直接使用它；实际中会依赖标准正态表或图形计算器（GDC）。
7. Standard Normal Distribution and Z‑tables | 标准正态分布与 Z 表
The standard normal distribution Z ~ N(0, 1²) has mean 0 and standard deviation 1. Its cumulative distribution function Φ(z) = P(Z ≤ z) is tabulated for positive z‑scores; the IB statistical table provides Φ(z) to four decimal places. A typical layout looks up the first two digits in the left column and the third decimal place across the top row.
标准正态分布 Z ~ N(0, 1²) 的均值为 0，标准差为 1。其累积分布函数 Φ(z) = P(Z ≤ z) 针对正的 z 分数已制成表格；IB 统计表将 Φ(z) 提供到小数点后四位。典型的布局是左侧列查找 z 的前两位数字，顶行查找第三位小数。
Example – Small portion of the standard normal table:
示例 – 标准正态表的一小部分：
| z | 0.00 | 0.01 | 0.02 | 0.03 |
|---|---|---|---|---|
| 1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 |
| 1.1 | 0.8643 | 0.8665 | 0.8686 | 0.8708 |
To find P(Z ≤ 1.02), move along the row for 1.0 to the column headed 0.02; the entry is 0.8461. Because the curve is symmetric, P(Z ≤ –z) = 1 – Φ(z). For example, P(Z ≤ –1.02) = 1 – 0.8461 = 0.1539.
要查 P(Z ≤ 1.02)，沿着 1.0 行移动到列标题为 0.02 的位置，条目为 0.8461。由于曲线对称，P(Z ≤ –z) = 1 – Φ(z)。例如，P(Z ≤ –1.02) = 1 – 0.8461 = 0.1539。
For a general normal variable X ~ N(μ, σ²), first standardise: Z = (X – μ)/σ. Then use the table to find probabilities for Z, and translate back to the X scale. This technique appears in many IB problems.
对于一般的正态变量 X ~ N(μ, σ²)，先进行标准化：Z = (X – μ)/σ。然后利用表格求出 Z 的概率，再转换回 X 尺度。这一技巧出现在大量 IB 试题中。
8. Worked Examples with Tables | 结合表格的例题解析
Example 1 – Discrete distribution table
A discrete random variable Y has the following probability distribution:
| y | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| P(Y = y) | 0.2 | a | 0.3 | 0.1 |
(i) Find the value of a. (ii) Calculate E(Y) and Var(Y).
例题 1 – 离散分布表
离散随机变量 Y 的概率分布如上。（i）求 a 的值。（ii）计算 E(Y) 和 Var(Y)。
Solution: Since the probabilities sum to 1, 0.2 + a + 0.3 + 0.1 = 1 → a = 0.4. E(Y) = 2×0.2 + 4×0.4 + 6×0.3 + 8×0.1 = 0.4 + 1.6 + 1.8 + 0.8 = 4.6. E(Y²) = 2²×0.2 + 4²×0.4 + 6²×0.3 + 8²×0.1 = 0.8 + 6.4 + 10.8 + 6.4 = 24.4. Hence Var(Y) = 24.4 – (4.6)² = 24.4 – 21.16 = 3.24, σ = √3.24 = 1.8.
解答：因为概率之和为 1，0.2 + a + 0.3 + 0.1 = 1 → a = 0.4。E(Y) = 2×0.2 + 4×0.4 + 6×0.3 + 8×0.1 = 4.6。E(Y²) = 2²×0.2 + 4²×0.4 + 6²×0.3 + 8²×0.1 = 24.4。因此 Var(Y) = 24.4 – (4.6)² = 3.24，σ = 1.8。
Example 2 – Binomial table
In a factory, the probability that a bulb is defective is 0.05. A random sample of 10 bulbs is taken. Use the cumulative binomial table for n=10, p=0.05 to find: (a) P(X ≤ 1) (b) P(X = 1) (c) P(X ≥ 2). The table gives P(X ≤ 0)=0.5987, P(X ≤ 1)=0.9139, P(X ≤ 2)=0.9885.
例题 2 – 二项分布表
一家工厂的灯泡次品率为 0.05。随机抽取 10 只灯泡。使用 n=10, p=0.05 的累积二项分布表求：(a) P(X ≤ 1) (b) P(X = 1) (c) P(X ≥ 2)。表提供 P(X ≤ 0)=0.5987, P(X ≤ 1)=0.9139, P(X ≤ 2)=0.9885。
Solution: (a) P(X ≤ 1) = 0.9139 directly from the table. (b) P(X = 1) = P(X ≤ 1) – P(X ≤ 0) = 0.9139 – 0.5987 = 0.3152. (c) P(X ≥ 2) = 1 – P(X ≤ 1) = 1 – 0.9139 = 0.0861.
解答：(a) P(X ≤ 1) = 0.9139 可直接查表。(b) P(X = 1) = P(X ≤ 1) – P(X ≤ 0) = 0.3152。(c) P(X ≥ 2) = 1 – P(X ≤ 1) = 0.0861。
Example 3 – Normal distribution
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