📚 Mathematics HL Option: Statistics & Probability | IB数学HL选修：统计与概率
The IB Mathematics HL Option on Statistics & Probability builds a rigorous foundation in both probability theory and statistical inference. It extends core concepts to include probability generating functions, continuous and bivariate distributions, the central limit theorem, estimation, and hypothesis testing. Mastering this option equips students with the tools to model uncertainty, analyse data, and draw reliable conclusions – skills essential for further study in mathematics, economics, and the sciences.
IB 数学 HL 的统计与概率选修课程为概率论与统计推断奠定了严谨的基础。它拓展了核心概念，涵盖概率生成函数、连续分布与二元分布、中心极限定理、参数估计和假设检验等内容。掌握该选修内容，学生将具备建模不确定性、分析数据并得出可靠结论的能力——这些技能对数学、经济学以及自然科学领域的深造至关重要。
1. Discrete Random Variables & Expectation | 离散随机变量与期望
mass function P(X = x), where the sum of probabilities over all possible values equals 1.
随机变量为概率试验的每一个结果赋予一个实数。若它只能取可数个值（例如整数），则称为离散随机变量。其分布由概率质量函数 P(X = x) 定义，且所有可能取值的概率之和为 1。
The expectation (or mean) of a discrete X is a measure of central location:
E(X) = μ = Σ x·P(X = x)
离散随机变量 X 的期望（均值）衡量其中心位置：
E(X) = μ = Σ x·P(X = x)
erty simplifies many calculations, especially when combining independent or scaled random variables.
期望具有线性性质：对于常数 a 和 b，E(aX + b) = aE(X) + b。这一性质可简化大量计算，尤其当对随机变量进行放缩或组合时更为方便。
For instance, a fair six‑ cted value is not necessarily a value that X can actually take, but it represents the long‑run average.
例如，一枚均匀六面骰子的期望值为 E(X) = (1+2+3+4+5+6)/6 = 3.5。期望值不一定在随机变量的取值范围内，但它代表长期重复试验的平均结果。
2. Variance, Covariance & Correlation | 方差、协方差与相关系数
Variance quantifies the dispersion of a random variable around its mean:
Var(X) = E[(X – μ)²] = E(X²) – μ²
方差度量随机变量在其均值周围的离散程度：
Var(X) = E[(X – μ)²] = E(X²) – μ²
onstants a and b, Var(aX + b) = a² Var(X). The additive constant b does not affect spread.
标准差 σ = √Var(X) 与 X 同单位，通常更易理解。对于常数 a 和 b，Var(aX + b) = a² Var(X)，可见加性常数 b 不影响离散程度。
Covariance measures the joint variability of two random variables:
Cov(X, Y) = E[(X – μₓ)(Y – μᵧ)] = E(XY) – μₓ μᵧ
协方差度量两个随机变量的联合变异程度：
Cov(X, Y) = E[(X – μₓ)(Y – μᵧ)] = E(XY) – μₓ μᵧ
The Pearson correlation coefficient standardises covariance to a scale‑free number between –1 and 1:
ρ = Cov(X, Y) / (σₓ σᵧ)
皮尔逊相关系数将协方差标准化为介于 –1 到 1 之间的无量纲数：
ρ = Cov(X, Y) / (σₓ σᵧ)
d Y are independent, then Cov(X, Y) = 0 and ρ = 0, but the converse is not generally true.
若 X 与 Y 独立，则 Cov(X, Y) = 0 且 ρ = 0，但逆命题不恒成立。
3. Probability Generating Functions | 概率生成函数
For a discrete random variable X taking non‑negative integer values, the probability generating function (PGF) is defined as
Gₓ(t) = E(tˣ) = Σ tˣ P(X = x)
对于取非负整数值的离散随机变量 X，其概率生成函数（PGF）定义为
Gₓ(t) = E(tˣ) = Σ tˣ P(X = x)
The series converges absolutely for |t| ≤ 1. PGFs possess powerful properties: differentiating at t = 1 gives G'ₓ(1) = E(X) and G''ₓ(1) = E[X(X – 1)], from which Var(X) can be derived.
该级数在 |t| ≤ 1 时绝对收敛。概率生成函数具有强大的性质：在 t = 1 处求导可得 G'ₓ(1) = E(X) 以及 G''ₓ(1) = E[X(X – 1)]，进而推导出 Var(X)。
t) Gᵧ(t). This makes the PGF especially useful when working with sums of independent random variables, such as the binomial distribution (which can be viewed as a sum of Bernoulli trials).
若 X 与 Y 独立，则它们之和的 PGF 为乘积：Gₓ₊ᵧ(t) = Gₓ(t) Gᵧ(t)。因此，在处理独立随机变量之和时（例如可将二项分布视作伯努利试验之和），PGF 尤为有用。
For a binomial distribution X ~ B(n, p), the PGF is Gₓ(t) = (q + pt)ⁿ, where q = 1 – p. Differentiating yields E(X) = np and Var(X) = npq.
对于二项分布 X ~ B(n, p)，其 PGF 为 Gₓ(t) = (q + pt)ⁿ，其中 q = 1 – p。求导可得 E(X) = np 及 Var(X) = npq。
4. Continuous Random Variables & PDF | 连续随机变量与概率密度函数
1 over the domain. Probabilities are obtained by integration: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
连续随机变量可在某区间内取任意值。其分布由概率密度函数（PDF）f(x) 描述，满足 f(x) ≥ 0 且在定义域上 ∫ f(x) dx = 1。概率通过积分求得：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。
The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫₋∞ˣ dt. The median m satisfies F(m) = 0.5, and the mode is the value of x that maximises f(x).
累积分布函数（CDF）为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。中位数 m 满足 F(m) = 0.5，众数则是使 f(x) 最大的 x 值。
Expectation and variance are defined analogously using integrals:
E(X) = ∫ x f(x) dx, Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ²
期望与方差用积分相似定义：
E(X) = ∫ x f(x) dx, Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ²
For a continuous uniform distribution on [a, b], f(x) = 1/(b – a), E(X) = (a + b)/2, and Var(X) = (b – a)²/12.
对于区间 [a, b] 上的连续均匀分布，f(x) = 1/(b – a)，E(X) = (a + b)/2，Var(X) = (b – a)²/12。
5. Normal Distribution & Central Limit Theorem | 正态分布与中心极限定理
The normal distribution with mean μ and variance σ², denoted X ~ N(μ, σ²), has PDF
f(x) = (1/(σ√2π)) e–½((x–μ)/σ)²
均值为 μ、方差为 σ² 的正态分布记作 X ~ N(μ, σ²)，其概率密度函数为
f(x) = (1/(σ√2π)) e–½((x–μ)/σ)²
Standardising gives Z = (X – μ)/σ ~ N(0, 1), which allows the use of standard normal tables.
标准化后 Z = (X – μ)/σ ~ N(0, 1)，方便使用标准正态表。
ndom sample of size n from any distribution with mean μ and variance σ², the sample mean X̄ is approximately normally distributed: X̄ ~ N(μ, σ²/n), provided n is sufficiently large (usually n ≥ 30).
中心极限定理（CLT）指出，对于从任意均值为 μ、方差为 σ² 的总体中抽取的容量为 n 的随机样本，只要 n 足够大（通常 n ≥ 30），样本均值 X̄ 近似服从正态分布：X̄ ~ N(μ, σ²/n)。
The CLT underpins many statistical procedures, allowing us to construct confidence intervals and conduct hypothesis tests even when the population distribution is unknown.
中心极限定理是众多统计方法的基础，即便总体分布未知，我们依然可以构建置信区间并进行假设检验。
6. Sampling Distributions & t‑Distribution | 抽样分布与t分布
A sampling distribution is the probability distribution of a sample statistic (such as the mean). If the population is normal and σ is known, the sample mean follows exactly X̄ ~ N(μ, σ²/n).
抽样分布是样本统计量（如样本均值）的概率分布。若总体服从正态且 σ 已知，则样本均值精确服从 X̄ ~ N(μ, σ²/n)。
is replaced by the sample standard deviation s. The resulting statistic follows a Student's t‑distribution with ν = n – 1 degrees of freedom:
T = (X̄ – μ) / (s/√n) ~ tn–1
当总体标准差 σ 未知时，用样本标准差 s 代替，得到统计量服从自由度为 ν = n – 1 的 t 分布：
T = (X̄ – μ) / (s/√n) ~ tn–1
The t‑distribution is symmetric and bell‑shaped but has heavier tails than the normal distribution, especially for small n. As n increases, it approaches the standard normal distribution.
t 分布对称且呈钟形，但尾部比正态分布更厚，尤其在样本量较小时。随着 n 增大，t 分布趋近于标准正态分布。
The sample variance s² = (1/(n–1)) Σ (xᵢ – X̄)² is an unbiased estimator of σ², which is why the t‑statistic uses this divisor.
样本方差 s² = (1/(n–1)) Σ (xᵢ – X̄)² 是 σ² 的无偏估计量，因此 t 统计量使用该分母。
7. Point & Interval Estimation | 点估计与区间估计
An estimator θ̂ is a statistic used to infer a population parameter θ. It is unbiased if E(θ̂) = θ. For example, the sample mean X̄ is an unbiased estimator of μ, and the sample variance s² is unbiased for σ².
估计量 θ̂ 是用于推断总体参数 θ 的统计量。若满足 E(θ̂) = θ，则它是无偏的。例如，样本均值 X̄ 是 μ 的无偏估计量，样本方差 s² 是 σ² 的无偏估计量。
A confidence interval gives a range of plausible values for θ. For a normal population with known σ, a 95% confidence interval for μ is
X̄ ± z0.025 · (σ/√n)
置信区间给出了参数 θ 的合理范围。对于 σ 已知的正态总体，μ 的 95% 置信区间为
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