剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

概率、随机变量与概率分布

AP 统计学 · 第 4 主题

训练
讲义 词汇表
4.1

统计学导论：随机与非随机模式？
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-1
Given that variation may be random or not, conclusions are uncertain.

VAR-1.F
Identify questions suggested by patterns in data. [Skill 1.A]

  • VAR-1.F.1 Patterns in data do not necessarily mean that variation is not random.

来源:美国大学理事会 AP 课程与考试说明

某件事是随机(random)的,若个体结果不确定但在许多次重复上出现一个规则的模式。短期结果看起来无规律;长期相对频率稳定下来。这个长期稳定性正是使概率有用的东西。

词汇表 训练
英文 中文 拼音
random 随机 suí jī
练习卷
4.2

用模拟估计概率
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-2
Simulation allows us to anticipate patterns in data.

UNC-2.A
Estimate probabilities using simulation. [Skill 3.A]

  • UNC-2.A.1 A random process generates results that are determined by chance.
  • UNC-2.A.2 An outcome is the result of a trial of a random process.
  • UNC-2.A.3 An event is a collection of outcomes.
  • UNC-2.A.4 Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. All possible outcomes are associated with a value to be determined by chance. Record the counts of simulated outcomes and the count total.
  • UNC-2.A.5 The relative frequency of an outcome or event in simulated or empirical data can be used to estimate the probability of that outcome or event.
  • UNC-2.A.6 The law of large numbers states that simulated (empirical) probabilities tend to get closer to the true probability as the number of trials increases.
    • Illustrative examples for UNC-2.A:
      • An outcome: Rolling a particular value on a six-sided number cube is one of six possible outcomes.
      • An event: When rolling two six-sided number cubes, an event would be a sum of seven. The corresponding collection of outcomes would be $(1, 6)$, $(2, 5)$, $(3, 4)$, $(4, 3)$, $(5, 2)$, and $(6, 1)$, where the ordered pairs indicate (face value on one cube, face value on the other cube).

来源:美国大学理事会 AP 课程与考试说明

一个模拟(simulation)用随机数字或技术模仿一个机会过程。步骤:陈述模型、把数字分配给结果、运行许多次试验,并记录满足条件的试验的比例。所得的比例估计这个概率——更多试验给一个更好的估计。

词汇表 训练
英文 中文 拼音
simulation 模拟 mó nǐ
练习卷
4.3

概率导论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.A
Calculate probabilities for events and their complements. [Skill 3.A]

  • VAR-4.A.1 The sample space of a random process is the set of all possible non-overlapping outcomes.
  • VAR-4.A.2 If all outcomes in the sample space are equally likely, then the probability an event E will occur is defined as the fraction: $\dfrac{\text{number of outcomes in event E}}{\text{total number of outcomes in sample space}}$
  • VAR-4.A.3 The probability of an event is a number between 0 and 1, inclusive.
  • VAR-4.A.4 The probability of the complement of an event E, $E'$ or $E^{C}$, (i.e., not E) is equal to $1 - P(E)$.

VAR-4.B
Interpret probabilities for events. [Skill 4.B]

  • VAR-4.B.1 Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event will occur in the long run.

来源:美国大学理事会 AP 课程与考试说明

一个事件的概率(probability)是一个从 $0$ 到 $1$ 的数字,给出它的长期相对频率。样本空间(sample space)是所有结果的集合。对于一个事件 $A$,补(complement)法则:$P(A^c)=1-P(A)$。所有结果的概率求和为 $1$。

Probability runs from 0 (impossible) to 1 (certain)
概率从 0(不可能)到 1(确定)
探索

Explore probability with dice

Probability is the long-run fraction of times an outcome happens. Roll the dice many times and watch the experimental proportions settle toward the theoretical values.

词汇表 训练
英文 中文 拼音
probability 概率 gài lǜ
sample space 样本空间 yàng běn kōng jiān
complement 补 bǔ
练习卷
4.4

互斥事件
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.C
Explain why two events are (or are not) mutually exclusive. [Skill 4.B]

  • VAR-4.C.1 The probability that events $A$ and $B$ both will occur, sometimes called the joint probability, is the probability of the intersection of $A$ and $B$, denoted $P(A \cap B)$.
  • VAR-4.C.2 Two events are mutually exclusive or disjoint if they cannot occur at the same time. So $P(A \cap B) = 0$.

来源:美国大学理事会 AP 课程与考试说明

两个事件是互斥(mutually exclusive)(不相交)的,若它们不能同时发生。那么加法法则简化:

$$P(A\text{ or }B)=P(A)+P(B)\quad(\text{if mutually exclusive}).$$
一般地,$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$ ——减去重叠使它不被数两次。

A Venn diagram: the overlap is the intersection of two events
一个维恩图:重叠是两个事件的交集
词汇表 训练
英文 中文 拼音
mutually exclusive 互斥 hù chì
练习卷
4.5

条件概率
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.D
Calculate conditional probabilities. [Skill 3.A]

  • VAR-4.D.1 The probability that event $A$ will occur given that event $B$ has occurred is called a conditional probability and denoted $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$.
  • VAR-4.D.2 The multiplication rule states that the probability that events $A$ and $B$ both will occur is equal to the probability that event $A$ will occur multiplied by the probability that event $B$ will occur, given that $A$ has occurred. This is denoted $P(A \cap B) = P(A) \cdot P(B \mid A)$.

来源:美国大学理事会 AP 课程与考试说明

条件概率

$A$ 给定 $B$ 的条件概率(conditional probability)是

$$P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}.$$
它是一旦你知道 $B$ 发生了 $A$ 的机会。双向表使这些容易:限制到 $B$ 的行/列,然后求 $A$ 的份额。

On a tree diagram, multiply the probabilities along the branches
在一个树状图上,沿分支相乘概率
探索

Update a probability on new information

Conditional probability $P(B\mid A)$ is the chance of $B$ once you know $A$ happened. Change the branch probabilities and watch how conditioning reshapes the outcome.

词汇表 训练
英文 中文 拼音
conditional probability 条件概率 tiáo jiàn gài lǜ
练习卷
4.6

独立事件与事件的并
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-4
The likelihood of a random event can be quantified.

VAR-4.E
Calculate probabilities for independent events and for the union of two events. [Skill 3.A]

  • VAR-4.E.1 Events $A$ and $B$ are independent if, and only if, knowing whether event $A$ has occurred (or will occur) does not change the probability that event $B$ will occur.
  • VAR-4.E.2 If, and only if, events $A$ and $B$ are independent, then $P(A \mid B) = P(A)$, $P(B \mid A) = P(B)$, and $P(A \cap B) = P(A) \cdot P(B)$.
  • VAR-4.E.3 The probability that event $A$ or event $B$ (or both) will occur is the probability of the union of $A$ and $B$, denoted $P(A \cup B)$.
  • VAR-4.E.4 The addition rule states that the probability that event $A$ or event $B$ or both will occur is equal to the probability that event $A$ will occur plus the probability that event $B$ will occur minus the probability that both events $A$ and $B$ will occur. This is denoted $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

来源:美国大学理事会 AP 课程与考试说明

事件是独立(independent)的,若知道一个不改变另一个的概率:$P(A\mid B)=P(A)$。那么乘法法则简化:

$$P(A\text{ and }B)=P(A)\,P(B)\quad(\text{if independent}).$$
独立不等同于互斥——有非零概率的互斥事件实际上是相依(dependent)的(若一个发生,另一个不能)。

A sample space diagram lists every equally likely outcome
一个样本空间图列出每个同等可能的结果
探索

Combine events with a Venn diagram

For a union $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ — you subtract the overlap so it isn't counted twice. Switch the operation to see each region light up.

词汇表 训练
英文 中文 拼音
independent 独立 dú lì
练习卷
4.7

随机变量与概率分布导论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.A
Represent the probability distribution for a discrete random variable. [Skill 2.B]

  • VAR-5.A.1 The values of a random variable are the numerical outcomes of random behavior.
  • VAR-5.A.2 A discrete random variable is a variable that can only take a countable number of values. Each value has a probability associated with it. The sum of the probabilities over all of the possible values must be 1.
  • VAR-5.A.3 A probability distribution can be represented as a graph, table, or function showing the probabilities associated with values of a random variable.
  • VAR-5.A.4 A cumulative probability distribution can be represented as a table or function showing the probability of being less than or equal to each value of the random variable.
    • Illustrative examples for VAR-5.A: Outcomes of trials of a random process:
      • The sum of the outcomes for rolling two dice
      • The number of puppies in a randomly selected litter for a certain breed of dog

VAR-5.B
Interpret a probability distribution. [Skill 4.B]

  • VAR-5.B.1 An interpretation of a probability distribution provides information about the shape, center, and spread of a population and allows one to make conclusions about the population of interest.

来源:美国大学理事会 AP 课程与考试说明

一个随机变量(random variable)给一个机会过程的每个结果分配一个数字。一个概率分布(probability distribution)列出每个可能的值连同它的概率(它们求和为 $1$)。一个分布能是离散的(一个值的表)或连续的(一个曲线下面积模型,像正态)。

词汇表 训练
英文 中文 拼音
random variable 随机变量 suí jī biàn liàng
probability distribution 概率分布 gài lǜ fēn bù
练习卷
4.8

随机变量的均值与标准差
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.C
Calculate parameters for a discrete random variable. [Skill 3.B]

  • VAR-5.C.1 A numerical value measuring a characteristic of a population or the distribution of a random variable is known as a parameter, which is a single, fixed value.
  • VAR-5.C.2 The mean, or expected value, for a discrete random variable $X$ is $\mu_X = \sum x_i \cdot P(x_i)$.
  • VAR-5.C.3 The standard deviation for a discrete random variable $X$ is $\sigma_X = \sqrt{\sum (x_i - \mu_x)^2 \cdot P(x_i)}$.

VAR-5.D
Interpret parameters for a discrete random variable. [Skill 4.B]

  • VAR-5.D.1 Parameters for a discrete random variable should be interpreted using appropriate units and within the context of a specific population.

来源:美国大学理事会 AP 课程与考试说明

一个离散随机变量的均值(期望值)(expected value)是概率加权的平均:

$$\mu_X=E(X)=\sum x_i\,P(x_i).$$
标准差 $\sigma_X=\sqrt{\sum (x_i-\mu_X)^2\,P(x_i)}$ 测量离均值的典型散布。期望值是长期平均结果,不是你在任何单次试验上期望的一个值。

Worked example. 一个游戏以概率 $0.2$ 付 $\$5$,而以概率 $0.8$ 花你 $\$1$(一个 $-1$ 结果)。期望值是

$$E(X)=5(0.2)+(-1)(0.8)=1-0.8=\$0.20,$$
所以在许多次玩上你平均每次玩赚约 $20$ 分,即使没有单次玩恰好给那个。

练习卷
4.9

组合随机变量
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

VAR-5
Probability distributions may be used to model variation in populations.

VAR-5.E
Calculate parameters for linear combinations of random variables. [Skill 3.B]

  • VAR-5.E.1 For random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$.
  • VAR-5.E.2 Two random variables are independent if knowing information about one of them does not change the probability distribution of the other.
  • VAR-5.E.3 For independent random variables $X$ and $Y$ and real numbers $a$ and $b$, the mean of $aX + bY$ is $a\mu_x + b\mu_y$, and the variance of $aX + bY$ is $a^2\sigma^2_x + b^2\sigma^2_y$.

VAR-5.F
Describe the effects of linear transformations of parameters of random variables. [Skill 3.C]

  • VAR-5.F.1 For $Y = a + bX$, the probability distribution of the transformed random variable, $Y$, has the same shape as the probability distribution for $X$, so long as $a > 0$ and $b > 0$. The mean of $Y$ is $\mu_y = a + b\mu_x$. The standard deviation of $Y$ is $\sigma_y = |b|\sigma_x$.

来源:美国大学理事会 AP 课程与考试说明

当你相加或相减随机变量时,均值相加:$\mu_{X\pm Y}=\mu_X\pm\mu_Y$。若 $X$ 和 $Y$ 独立,方差相加(即使相减时):

$$\sigma^2_{X\pm Y}=\sigma^2_X+\sigma^2_Y.$$
取平方根得标准差。还有,缩放:$\mu_{aX+b}=a\mu_X+b$ 而 $\sigma_{aX+b}=|a|\sigma_X$。

练习卷
4.10

二项分布导论
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.A
Estimate probabilities of binomial random variables using data from a simulation. [Skill 3.A]

  • UNC-3.A.1 A probability distribution can be constructed using the rules of probability or estimated with a simulation using random number generators.
  • UNC-3.A.2 A binomial random variable, $X$, counts the number of successes in $n$ repeated independent trials, each trial having two possible outcomes (success or failure), with the probability of success $p$ and the probability of failure $1 - p$.

UNC-3.B
Calculate probabilities for a binomial distribution. [Skill 3.A]

  • UNC-3.B.1 The probability that a binomial random variable, $X$, has exactly $x$ successes for $n$ independent trials, when the probability of success is $p$, is calculated as $P(X = x) = \binom{n}{x} p^x (1 - p)^{n-x}, x = 0, 1, 2, \ldots, n$. This is the binomial probability function.

来源:美国大学理事会 AP 课程与考试说明

二项分布

一个二项(binomial)情形(BINS):一个固定数目 $n$ 的独立(Independent)试验,每个有两个结果(成功/失败)和相同的成功概率 $p$。随机变量 $X=$ 成功的次数。它的概率:

$$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}.$$

The binomial distribution, with mean n times p
二项分布,均值 n 乘 p
探索

Shape a binomial distribution

A binomial distribution counts successes in $n$ independent trials each with probability $p$. Change $n$ and $p$ and watch the bars shift and spread.

词汇表 训练
英文 中文 拼音
mean (expected value) 期望值 qī wàng zhí
binomial 二项 èr xiàng
练习卷
4.11

二项分布的参数
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.C
Calculate parameters for a binomial distribution. [Skill 3.B]

  • UNC-3.C.1 If a random variable is binomial, its mean, $\mu_x$, is $np$ and its standard deviation, $\sigma_x$, is $\sqrt{np(1 - p)}$.

UNC-3.D
Interpret probabilities and parameters for a binomial distribution. [Skill 4.B]

  • UNC-3.D.1 Probabilities and parameters for a binomial distribution should be interpreted using appropriate units and within the context of a specific population or situation.

来源:美国大学理事会 AP 课程与考试说明

对于一个有 $n$ 次试验和成功概率 $p$ 的二项 $X$:

$$\mu_X=np,\qquad \sigma_X=\sqrt{np(1-p)}.$$
对"我们期望多少次成功,而它们变化多少"的问题用这些。

Worked example. 一个球员命中 $70\%$ 的罚球。在 $n=10$ 次投篮里,恰好 $8$ 次命中的概率是

$$P(X=8)=\binom{10}{8}(0.7)^8(0.3)^2=45\times0.0576\times0.09\approx0.23,$$
而命中的期望数是 $\mu=np=10(0.7)=7$,带 $\sigma=\sqrt{10(0.7)(0.3)}\approx1.45$。

练习卷
4.12

几何分布
大纲
Enduring UnderstandingLearning ObjectiveEssential Knowledge

UNC-3
Probabilistic reasoning allows us to anticipate patterns in data.

UNC-3.E
Calculate probabilities for geometric random variables. [Skill 3.A]

  • UNC-3.E.1 For a sequence of independent trials, a geometric random variable, $X$, gives the number of the trial on which the first success occurs. Each trial has two possible outcomes (success or failure) with the probability of success $p$ and the probability of failure $1 - p$.
  • UNC-3.E.2 The probability that the first success for repeated independent trials with probability of success $p$ occurs on trial $x$ is calculated as $P(X = x) = (1 - p)^{x-1} p, x = 1, 2, 3, \ldots$. This is the geometric probability function.

UNC-3.F
Calculate parameters of a geometric distribution. [Skill 3.B]

  • UNC-3.F.1 If a random variable is geometric, its mean, $\mu_x$, is $\dfrac{1}{p}$ and its standard deviation, $\sigma_x$, is $\dfrac{\sqrt{(1 - p)}}{p}$.

UNC-3.G
Interpret probabilities and parameters for a geometric distribution. [Skill 4.B]

  • UNC-3.G.1 Probabilities and parameters for a geometric distribution should be interpreted using appropriate units and within the context of a specific population or situation.

来源:美国大学理事会 AP 课程与考试说明

一个几何(geometric)情形与二项相同但没有固定的 $n$:你持续尝试直到第一次成功。随机变量 $Y=$ 第一次成功的试验:

$$P(Y=k)=(1-p)^{k-1}\,p,\qquad \mu_Y=\frac{1}{p}.$$
所以到第一次成功的试验的期望数是 $1/p$。

词汇表 训练
英文 中文 拼音
geometric 几何 jǐ hé
练习卷
4.12

考试技巧
  • 一个概率落在 $[0,1]$ 里;用补($1-P$)并相加互斥事件。
  • 对独立事件相乘;对"与/或"用一般的加法和条件法则。
  • 期望值 = $\sum(\text{value}\times\text{probability})$。
  • 辨认二项(固定 $n$、两个结果、恒定 $p$)和几何情形。
  • 对多阶段问题画一个树或表并沿分支相乘。

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