Probability and Statistics Foundations概率与统计基础

• Sample space $S$.样本空间 $S$。 The set of all possible outcomes of an experiment. Coin: $S = \{H, T\}$. Die: $S = \{1, 2, 3, 4, 5, 6\}$. Two coins: $S = \{HH, HT, TH, TT\}$.实验所有可能结果的集合。掷一枚硬币：$S = \{H, T\}$；掷一颗骰子：$S = \{1, 2, 3, 4, 5, 6\}$；掷两枚硬币：$S = \{HH, HT, TH, TT\}$。
• Event $A$.事件 $A$。 A subset of $S$. "Roll an even number" is the event $A = \{2, 4, 6\} \subseteq S$. CCSSM HSS-CP.A.1 names this verbatim.$S$ 的子集。"掷出偶数"即事件 $A = \{2, 4, 6\} \subseteq S$。CCSSM HSS-CP.A.1 原文如此。
• Classical formula经典公式 (equally likely outcomes only):（仅适用等可能结果）：

• Complement rule.补集规则。 $P(A^c) = 1 - P(A)$. Often easier: "at least one" $=$ $1 - $ "none".$P(A^c) = 1 - P(A)$。常用技巧："至少一个" $=$ $1 - $"一个也没有"。
• Addition rule.加法规则。 $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Subtract the overlap so you don't count it twice. CCSSM HSS-CP.B.7.$P(A \cup B) = P(A) + P(B) - P(A \cap B)$。减去交集避免重复计数。CCSSM HSS-CP.B.7。
• Range.取值范围。 $0 \le P(A) \le 1$. $P(S) = 1$ (something must happen); $P(\emptyset) = 0$ (impossible event).$0 \le P(A) \le 1$。$P(S) = 1$（必然事件）；$P(\emptyset) = 0$（不可能事件）。

Conditional probability:条件概率：

Multiplication rule:乘法规则：

Bayes' rule (the conditional probabilities $P(A | B)$ and $P(B | A)$ are exchanged):贝叶斯法则（交换条件概率 $P(A | B)$ 与 $P(B | A)$）：

• Law of total probability.全概率公式。 If $A_1, \ldots, A_n$ partition $S$ then $P(B) = \sum_i P(B | A_i) P(A_i)$. This is the denominator of Bayes' rule when you have several "hypotheses" $A_i$.若 $A_1, \ldots, A_n$ 是 $S$ 的一个划分，则 $P(B) = \sum_i P(B | A_i) P(A_i)$。当有多个"假设" $A_i$ 时，这是贝叶斯法则的分母。
• Two-way table reading.双向表读取。 $P(A | B)$ is the row (or column) relative frequency , restrict to the $B$ slice, then ask what fraction is also in $A$. CCSSM HSS-CP.B.6 writes this method verbatim.$P(A | B)$ 即"行（或列）相对频率", 先限制在 $B$ 这一切片内，再问其中有多少属于 $A$。CCSSM HSS-CP.B.6 原文表述此法。

• Equivalent forms.等价形式。 $P(A | B) = P(A)$ and $P(B | A) = P(B)$ , "knowing $B$ tells you nothing extra about $A$." CCSSM HSS-CP.A.2, .A.3.$P(A | B) = P(A)$ 与 $P(B | A) = P(B)$ , "知道 $B$ 不增添关于 $A$ 的信息"。CCSSM HSS-CP.A.2、.A.3。
• Independent vs mutually exclusive.独立 与 互斥。 Different concepts. Mutually exclusive: $A \cap B = \emptyset$, so $P(A \cap B) = 0$. Independent: $P(A \cap B) = P(A) P(B)$, generally nonzero. Two events with positive probability cannot be both.两个不同的概念。互斥：$A \cap B = \emptyset$，故 $P(A \cap B) = 0$；独立：$P(A \cap B) = P(A) P(B)$，通常不为零。两个概率均为正的事件不可能既互斥又独立。
• Sequential / dependent.序列 / 相依。 For without-replacement draws, the second probability depends on the first , use the multiplication rule with the conditional: $P(A_1 \cap A_2) = P(A_1) \, P(A_2 | A_1)$.无放回抽取时，第二次概率依赖第一次 , 用带条件的乘法规则：$P(A_1 \cap A_2) = P(A_1) \, P(A_2 | A_1)$。
• Tree diagrams.树图。 Multiply along a branch (intersections), add across branches (unions). AB Math 30-2 indicator 3.4 expects "determining the probability of dependent or independent events" via this kind of organizer.沿一条枝相乘（交集），跨枝相加（并集）。AB Math 30-2 指标 3.4 要求"判定相依或独立事件的概率"，正是用此类图示。

A discrete random variable $X$ takes finitely many values $x_1, x_2, \ldots, x_n$ with probabilities $p_i = P(X = x_i)$. The pair-list $\{(x_i, p_i)\}$ is the probability mass function (PMF). Required: $p_i \ge 0$ and $\sum p_i = 1$.离散随机变量 $X$ 取有限多个值 $x_1, x_2, \ldots, x_n$，对应概率 $p_i = P(X = x_i)$。对子列表 $\{(x_i, p_i)\}$ 即概率质量函数（PMF）。条件：$p_i \ge 0$ 且 $\sum p_i = 1$。

Expected value (mean):期望（均值）：

Variance and standard deviation:方差与标准差：

Computational shortcut:计算捷径：

• $E$ is a weighted average.$E$ 是加权平均。 Each value times its probability. If outcomes are equally likely $p_i = 1/n$, then $E(X)$ reduces to the ordinary mean.每个值乘以其概率。等可能时 $p_i = 1/n$，$E(X)$ 退化为普通平均。
• Variance has squared units.方差单位带平方。 If $X$ is in metres, $\mathrm{Var}(X)$ is in metres-squared; $\sigma$ restores the original units. Standard deviation is usually the reportable spread.若 $X$ 单位为米，$\mathrm{Var}(X)$ 单位为米-平方；$\sigma$ 回到原单位。标准差通常是可报告的离散度。
• Linearity.线性性。 $E(aX + b) = a E(X) + b$ and $\mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)$ , constants come out of $E$, square out of $\mathrm{Var}$.$E(aX + b) = a E(X) + b$，$\mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)$ , 常数可从 $E$ 中提出，方差中需平方。

$X \sim B(n, p)$ means $X$ counts successes in $n$ trials where:$X \sim B(n, p)$ 表示 $X$ 计数 $n$ 次试验中的"成功"次数，且满足：

1. $n$ trials,$n$ 次试验， fixed in advance.事先固定。
2. Each trial is "success" or "failure"每次试验只有"成功"或"失败" (Bernoulli).（伯努利试验）。
3. $P(\text{success}) = p$ is the same on every trial.每次 $P($ 成功 $) = p$ 不变。
4. Trials are independent.各次试验独立。

PMF:概率质量函数：

Mean and variance:均值与方差：

• "At least", "at most" require sums."至少"、"至多"需要求和。 $P(X \ge k) = \sum_{j=k}^{n} P(X = j)$. Often easier via the complement: $P(X \ge 1) = 1 - P(X = 0) = 1 - (1-p)^n$.$P(X \ge k) = \sum_{j=k}^{n} P(X = j)$。常通过补集更省事：$P(X \ge 1) = 1 - P(X = 0) = 1 - (1-p)^n$。
• Why $E(X) = np$.$E(X) = np$ 的来由。 Write $X = X_1 + X_2 + \cdots + X_n$ where $X_i$ indicates "success on trial $i$" ($X_i = 1$ or $0$). Then $E(X_i) = p$ and $E(X) = \sum E(X_i) = np$ by linearity. Independence is not needed for the mean , it is needed for the variance.写 $X = X_1 + X_2 + \cdots + X_n$，其中 $X_i$ 指示"第 $i$ 次成功"（$X_i = 1$ 或 $0$）。则 $E(X_i) = p$，由线性性 $E(X) = \sum E(X_i) = np$。均值不需独立性；方差需要。

• Shape and parameters.形状与参数。 A normal distribution $N(\mu, \sigma^2)$ is a symmetric bell-shaped curve centred at $\mu$ (mean = median = mode) with spread controlled by $\sigma$. Total area under the curve is $1$.正态分布 $N(\mu, \sigma^2)$ 以 $\mu$ 为对称中心呈钟形（均值 = 中位数 = 众数），离散度由 $\sigma$ 控制。曲线下总面积 $= 1$。
• $68$-$95$-$99.7$ rule (empirical rule).$68$-$95$-$99.7$ 法则（经验法则）。 $P(|X - \mu| \le \sigma) \approx 0.68$, $P(|X - \mu| \le 2\sigma) \approx 0.95$, $P(|X - \mu| \le 3\sigma) \approx 0.997$. About $2/3$ within $1\sigma$, $95\%$ within $2\sigma$, virtually all within $3\sigma$.$P(|X - \mu| \le \sigma) \approx 0.68$；$P(|X - \mu| \le 2\sigma) \approx 0.95$；$P(|X - \mu| \le 3\sigma) \approx 0.997$。约 $2/3$ 在 $\pm 1\sigma$、$95\%$ 在 $\pm 2\sigma$、几乎全部在 $\pm 3\sigma$。
• $z$-score.$z$-分数。

$z$ is the number of standard deviations $x$ is above ($z > 0$) or below ($z < 0$) the mean. Standardising lets one $z$-table serve every normal distribution.$z$ 表示 $x$ 高于（$z > 0$）或低于（$z < 0$）均值多少个标准差。标准化后一张 $z$-表通用于所有正态分布。

• Standard normal.标准正态。 $Z \sim N(0, 1)$. $P(Z \le z)$ is read from a $z$-table or calculator (e.g. normalcdf).$Z \sim N(0, 1)$。$P(Z \le z)$ 由 $z$-表或计算器（如 normalcdf）读取。
• Symmetry.对称性。 $P(Z \le -z) = 1 - P(Z \le z)$ , lets you handle negatives with a positive table.$P(Z \le -z) = 1 - P(Z \le z)$ , 用正值表即可处理负值。

• Mean均值 $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$. Sensitive to outliers.对异常值敏感。
• Median.中位数。 Sort, take the middle ($n$ odd) or the average of the two middles ($n$ even). Resistant to outliers , use median when the data is skewed.排序后取中间（$n$ 奇）或两中间值的平均（$n$ 偶）。对异常值稳健 , 数据偏态时用中位数。
• Mode.众数。 Most-frequent value. Can be none, one, or multiple. Most useful for categorical or discrete data.最高频值。可无、可一、可多。最适合分类或离散数据。
• Range极差 $= x_{\max} - x_{\min}$. Simplest spread; uses only two values, so extremely sensitive to outliers.最简单的离散度量；只用两个值，对异常值极度敏感。
• IQRIQR（四分位距） $= Q_3 - Q_1$. Resistant spread , the middle $50\%$ of the data. CCSSM HSS-ID.A.3: "interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers)."稳健离散度 , 中间 $50\%$ 数据。CCSSM HSS-ID.A.3："在数据集背景下解释形状、中心、离散度的差异，并考虑极端数据点（异常值）的可能影响"。

Standard deviation (sample):标准差（样本）：

Use $n$ in the denominator for a population, $n - 1$ for a sample (Bessel's correction , the curriculum usually says "sample standard deviation" or assumes technology).总体用分母 $n$；样本用 $n - 1$（贝塞尔修正 , 课纲通常写"样本标准差"或假设用技术工具）。

• Boxplot (five-number summary):箱线图（五数概括）： min, $Q_1$, median, $Q_3$, max. Box from $Q_1$ to $Q_3$ (length $=$ IQR); whiskers to min and max (or to inside $1.5 \cdot $ IQR, with outliers shown as dots).最小值、$Q_1$、中位数、$Q_3$、最大值。箱体从 $Q_1$ 到 $Q_3$（长度 $=$ IQR）；须延伸至最大 / 最小（或 $1.5 \cdot$ IQR 内，异常值另点出）。
• Histogram.直方图。 Bars over equal-width bins; height shows frequency or density. Shape labels: symmetric, right-skewed (tail to the right, mean $>$ median), left-skewed.等宽箱内的柱形；高度为频数或密度。形状标签：对称、右偏（尾在右，均值 $>$ 中位数）、左偏。
• Comparison rule.比较规则。 CCSSM HSS-ID.A.2: "use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets." If data is skewed, compare median + IQR; if symmetric, mean + standard deviation.CCSSM HSS-ID.A.2："使用与数据分布形状相匹配的统计量，比较两个或多个数据集的中心与离散度"。偏态时比较中位数 + IQR；对称时比较均值 + 标准差。