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概率、随机变量与概率分布

AP 统计学 · 第 4 主题

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Probability, Random Variables, and Probability Distributions

A basketball player makes five shots in a row. The crowd is sure she is hot tonight. But watch what chance alone can do. Flip a fair coin twenty times. There…

英文讲解 · 内嵌中英文字幕

讲义 双页 词汇表 词汇测验 3 双页 词汇测验 4 双页
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/ˈrændəm/ 随机 suí jī
练习卷 双页
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/ˌsɪmjʊˈleɪʃn/ 模拟 mó nǐ
练习卷 双页
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/ˌprɒbəˈbɪlɪti/ 概率 gài lǜ
sample space/ˈsæmpl speɪs/ 样本空间 yàng běn kōng jiān
complement/ˈkɒmplɪmənt/ 补 bǔ
练习卷 双页
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/ˈmjuːtʃuːəli eksˈkluːsɪv/ 互斥 hù chì
练习卷 双页
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/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ 条件概率 tiáo jiàn gài lǜ
练习卷 双页
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/ˌɪndɪˈpendənt/ 独立 dú lì
练习卷 双页
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/ˈrændəm ˈveərɪəbl/ 随机变量 suí jī biàn liàng
probability distribution/ˌprɒbəˈbɪlɪti ˌdɪstrɪˈbjuːʃn/ 概率分布 gài lǜ fēn bù
练习卷 双页
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 双页
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 双页
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)/miːn/ 期望值 qī wàng zhí
binomial/baɪˈnəʊmɪəl/ 二项 èr xiàng
练习卷 双页
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 双页
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/ˌdʒiːəʊˈmetrɪk/ 几何 jǐ hé
练习卷 双页
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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