📚 Review Set 8C: Probability Distributions & Hypothesis Testing | 复习卷8C：概率分布与假设检验
Welcome to Review Set 8C, designed to help IB Mathematics students consolidate their understanding of probability distributions and inferential statistics. This set focuses on discrete and continuous distributions, sampling, confidence intervals, and hypothesis testing—vital skills for both Analysis & Approaches and Applications & Interpretation courses. Work through these reminders to strengthen your exam readiness.
欢迎使用复习卷8C，本卷旨在帮助IB数学学生巩固对概率分布和推断统计学的理解。重点涵盖离散与连续分布、抽样、置信区间和假设检验，这些都是分析与方法以及应用与解释课程的核心技能。认真消化这些复习要点，为考试做好充分准备。
1. Discrete Random Variables | 离散随机变量
n IB exams, you may be asked to find an unknown probability using this summation property.
离散随机变量只能取有限个或可数个不同的值。概率质量函数给每个可能取值 xᵢ 分配概率 pᵢ，且满足 Σpᵢ = 1。在IB考试中，你可能需要利用这一求和性质求未知概率。
The expected value, or mean, is E(X) = μ = Σ xᵢ pᵢ. It represents the theoretical long‑ the expected score is found by multiplying each face value by its probability and summing.
期望值即均值，计算公式为 E(X) = μ = Σ xᵢ pᵢ。它代表理论上的长期平均值。例如，若 X 表示一枚不均匀骰子的点数，期望得分可通过各点数乘以其概率再求和得到。
X²) − μ² is often quicker, especially when a probability distribution is given in a table.
方差 Var(X) = Σ (xᵢ − μ)² pᵢ 衡量分布的离散程度。快速公式 Var(X) = E(X²) − μ² 通常更高效，特别是当概率分布以表格形式给出时。
Below is a simple example of a DRV table:
下面是一个简单的离散随机变量表格示例：
| xᵢ | 1 | 2 | 3 |
|---|---|---|---|
| P(X = xᵢ) | 0.3 | 0.5 | 0.2 |
+ 3×0.2 = 1.9, and E(X²) = 1²×0.3 + 2²×0.5 + 3²×0.2 = 4.1, so Var(X) = 4.1 − 1.9² = 0.49.
于是 E(X) = 1×0.3 + 2×0.5 + 3×0.2 = 1.9，E(X²) = 1²×0.3 + 2²×0.5 + 3²×0.2 = 4.1，故 Var(X) = 4.1 − 1.9² = 0.49。
2. Linear Transformations and Sums | 线性变换与求和
a²Var(X). This is a favourite topic in Set 8C because it tests fundamental understanding.
当离散随机变量进行线性变换 Y = aX + b 时，期望和方差有规律地变化：E(Y) = aE(X) + b，Var(Y) = a²Var(X)。这是复习卷8C中的热门考点，因为它考验基本概念。
− Y) also equals Var(X) + Var(Y) because the variance of a difference adds the variances.
对于独立随机变量 X 和 Y，E(X + Y) = E(X) + E(Y) 且 Var(X + Y) = Var(X) + Var(Y)。注意 Var(X − Y) 也等于 Var(X) + Var(Y)，因为差的方差等于方差之和。
These properties are essential for building up distributions for repeated bets, games, or combined measurements.
这些性质对于构建重复投注、游戏或组合测量的分布至关重要。
3. Binomial Distribution | 二项分布
If X ~ B(n, p), the probability of exactly r successes is given by P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, where ⁿCᵣ = n!/(r!(n−r)! . These formulas are provided in the IB data booklet, but you must know how to apply them.
若 X ~ B(n, p)，恰好获得 r 次成功的概率由 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ 给出，其中 ⁿCᵣ = n!/(r!(n−r)!) 。均值为 np，方差为 np(1−p)。这些公式在IB公式手册中提供，但你需要知道如何应用。
The binomial distribution requires independent trials with a fixed probability of success p. Cumulative probabilities can be found using your calculator's binomial CDF function. Always check that conditions are satisfied: fixed number of trials, two outcomes per trial, constant probability, and independence.
二项分布要求各次试验相互独立且成功概率 p 固定不变。累积概率可使用计算器中的二项累积分布函数求得。务必检查条件是否满足：固定次数的试验、每次试验两个结果、概率恒定以及独立性。
When solving problems, be careful to distinguish between 'at least', 'at most', and 'exactly'. For example, P(X ≥ 3) = 1 − P(X ≤ 2) using the binomial CDF.
解题时要小心区分“至少”、“最多”和“恰好”。例如，P(X ≥ 3) = 1 − P(X ≤ 2)，可使用二项累积分布函数计算。
4. Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is a continuous bell-shaped curve symmetric about the mean μ. The total area under the curve is 1. Probabilities correspond to areas under the density function. The empirical rule states that about 68% of values lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
正态分布 N(μ, σ²) 是一条关于均值 μ 对称的连续钟形曲线。曲线下的总面积为 1。概率对应于密度函数曲线下方的面积。经验法则指出，约68%的数值落在μ的1σ范围内，95%在2σ范围内，99.7%在3σ范围内。
N(170, 25), the probability a person is taller than 180 cm is P(X > 180) = 1 − P(X < 180).
通常需要使用计算器的正态累积分布函数计算 P(a < X < b)。在许多问题中，身高、体重和考试成绩等现实变量都可以用正态分布建模。例如，如果身高服从 N(170, 25)，一个人高于180厘米的概率为 P(X > 180) = 1 − P(X < 180)。
5. Standardising to Z | 标准化与 Z 值
To compare or compute probabilities without technology, we standardise: Z = (X − μ)/σ ~ normal CDF. Tables of Φ were historically used; today calculators provide direct methods.
为了不使用技术工具进行比较或概率计算，我们进行标准化：Z = (X − μ)/σ ~ N(0, 1)。于是 P(X < x) = Φ(z)，其中 Φ 是标准正态累积分布函数。历史上使用 Φ 表，如今计算器提供直接计算方法。
z) = 0.975, then z ≈ 1.96. This critical value appears frequently in confidence intervals.
逆正态函数用于在给定概率下求 x 值，使得 P(X < x) = 给定概率。例如，若 P(Z < z) = 0.975，则 z ≈ 1.96。这一临界值经常出现在置信区间中。
6. Inverse Normal Calculations | 逆正态计算
erse normal function with parameters μ and σ to find x₀. Be careful with right-tail or two‑tail problems: draw a sketch and convert to left‑tail area. For example, to find the 90th percentile, solve P(X < x) = 0.90.
当已知 P(X < x₀) = p 时，使用带有参数
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