Probability Models and Their Applications | 概率模型及其应用

📚 Probability Models and Their Applications | 概率模型及其应用

Probability models are mathematical frameworks that describe random phenomena and quantify uncertainty. They are essential in statistics, science, engineering, finance, and everyday decision-making.

概率模型是描述随机现象并量化不确定性的数学框架。它们在统计、科学、工程、金融以及日常决策中至关重要。

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1. Random Experiments and Sample Spaces | 随机试验与样本空间

A random experiment is a procedure whose outcome cannot be predicted with certainty. The set of all possible outcomes is called the sample space, usually denoted by Ω (omega).

随机试验是结果无法确定预测的过程。所有可能结果的集合称为样本空间，通常用Ω表示。

For example, tossing a fair coin gives Ω = {H, T}. Rolling a fair six-sided die gives Ω = {1, 2, 3, 4, 5, 6}. Each outcome in a fair setting is equally likely.

例如，抛一枚公平的硬币得到Ω = {H, T}。掷一枚公平的六面骰子得到Ω = {1, 2, 3, 4, 5, 6}。在公平情况下，每个结果等可能发生。

An event is any subset of the sample space. Events can be combined using set operations:

事件是样本空间的任意子集。事件可以通过集合运算组合：

• A ∪ B means “A or B occurs” (union).
• A ∪ B表示”A或B发生”（并）。
• A ∩ B means “A and B occur” (intersection).
• A ∩ B表示”A与B都发生”（交）。
• Aᶜ means “A does not occur” (complement).
• Aᶜ表示”A不发生”（补）。

For equally likely outcomes, the probability of event A is P(A) = |A| / |Ω|, where |A| is the number of outcomes in A.

对于等可能结果，事件A的概率为P(A) = |A| / |Ω|，其中|A|是A中结果的个数。

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2. Probability Axioms and Basic Rules | 概率公理与基本规则

The three axioms of probability form the foundation of all probability models:

概率的三条公理构成了所有概率模型的基础：

• Axiom 1: For any event A, 0 ≤ P(A) ≤ 1.
• 公理1：对任何事件A，有0 ≤ P(A) ≤ 1。
• Axiom 2: The probability of the sample space is 1, i.e. P(Ω) = 1.
• 公理2：样本空间的概率为1，即P(Ω) = 1。
• Axiom 3: For mutually exclusive events A₁, A₂, …, P(A₁ ∪ A₂ ∪ …) = Σ P(Aᵢ).
• 公理3：对于互斥事件A₁、A₂、…，有P(A₁ ∪ A₂ ∪ …) = Σ P(Aᵢ)。

From these axioms we derive the addition rule:

由这些公理我们推导出加法规则：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B). Also, the complement rule states P(Aᶜ) = 1 − P(A).

对于互斥事件，P(A ∩ B) = 0，因此P(A ∪ B) = P(A) + P(B)。此外，补事件规则表明P(Aᶜ) = 1 − P(A)。

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3. Conditional Probability and Independence | 条件概率与独立性

Conditional probability measures the probability of an event A given that another event B has already occurred:

条件概率衡量在事件B已经发生的条件下事件A发生的概率：

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0

Rearranging gives the multiplication rule:

重新整理得到乘法规则：

P(A ∩ B) = P(A|B) × P(B)

Two events A and B are independent if P(A|B) = P(A), equivalently:

如果P(A|B) = P(A)，则事件A与B独立，等价于：

P(A ∩ B) = P(A) × P(B)

Independence means the occurrence of B does not change the probability of A.

独立性意味着B的发生不会改变A的概率。

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4. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem links a conditional probability to its inverse. Suppose events B₁, B₂, …, Bₙ form a partition of Ω, and A is any event. Then the law of total probability states:

贝叶斯定理将条件概率与其逆概率联系起来。假设事件B₁、B₂、…、Bₙ构成Ω的一个划分，且A是任意事件。那么全概率公式表明：

P(A) = Σ P(A|Bᵢ) P(Bᵢ)

Bayes’ theorem gives the posterior probability of Bᵢ given A:

贝叶斯定理给出在A发生条件下Bᵢ的后验概率：

P(Bᵢ|A) = [P(A|Bᵢ) P(Bᵢ)] / Σ P(A|Bⱼ) P(Bⱼ)

Example: A test for a disease is 99% accurate. If 1% of the population has the disease, Bayes’ theorem shows the probability that a person with a positive test actually has the disease is about 50%—surprisingly low, illustrating the importance of base rates.

例如：某项疾病检测的准确率为99%。如果人群中患病率为1%，贝叶斯定理显示检测呈阳性者实际患病的概率约为50%——出乎意料地低，这说明了基础概率的重要性。

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5. Discrete Random Variables | 离散随机变量

A discrete random variable X takes values in a countable set with a probability mass function (PMF), p(x) = P(X = x).

离散随机变量X在一个可数集合上取值，其概率质量函数（PMF）为p(x) = P(X = x)。

The conditions for a valid PMF are p(x) ≥ 0 and Σₓ p(x) = 1.

有效PMF的条件是p(x) ≥ 0且Σₓ p(x) = 1。

Example: Let X be the sum when two fair dice are rolled. The PMF can be displayed as:

例如：设X为掷两枚公平骰子的点数之和。其PMF可表示为：

x	2	3	4	5	6	7	8	9	10	11	12
P(X=x)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

The cumulative distribution function is F(x) = P(X ≤ x), and it increases stepwise for discrete variables.

累积分布函数为F(x) = P(X ≤ x)，对于离散变量它是阶梯式递增的。

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6. Expected Value and Variance | 期望与方差

The expected value (mean) of a discrete random variable is a weighted average of its possible values:

离散随机变量的期望（均值）是其可能取值的加权平均值：

E(X) = Σₓ x p(x)

The variance measures the spread of the distribution:

方差衡量分布的离散程度：

Var(X) = E(X²) − [E(X)]²

where E(X²) = Σₓ x² p(x). The standard deviation is the square root of the variance.

其中E(X²) = Σₓ x² p(x)。标准差是方差的正平方根。

Properties: For constants a and b, E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).

性质：对于常数a和b，有E(aX + b) = aE(X) + b，且Var(aX + b) = a² Var(X)。

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7. Binomial Distribution | 二项分布

The binomial model applies when there are n independent trials, each with two outcomes (success/failure) and constant success probability p. Let X be the number of successes.

二项模型适用于n次独立试验，每次试验只有两种结果（成功/失败），且成功概率p恒定。设X为成功次数。

The probability of exactly r successes is:

恰好r次成功的概率为：

P(X = r) = C(n, r) pʳ (1−p)ⁿ⁻ʳ

for r = 0, 1, …, n, where C(n, r) = n! / [r!(n−r)!].

其中r = 0, 1, …, n，且C(n, r) = n! / [r!(n−r)!]。

Its mean and variance are:

其均值和方差为：

E(X) = np, Var(X) = np(1−p)

Example: If 10 independent shots each have a 70% chance of hitting a target, the probability of exactly 8 hits is C(10,8)(0.7)⁸(0.3)² ≈ 0.233.

例如：如果10次独立射击每次命中目标的概率为70%，那么恰好命中8次的概率为C(10,8)(0.7)⁸(0.3)² ≈ 0.233。

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8. Poisson Distribution | 泊松分布

The Poisson distribution models the number of rare events occurring in a fixed interval of time or space. It has a single parameter λ, the average rate of occurrence.

泊松分布用于模拟在固定时间或空间间隔内稀有事件发生的次数。它有一个参数λ，即平均发生率。

The PMF is:

其PMF为：

P(X = r) = e^(−λ) λʳ / r!

for r = 0, 1, 2, …, with λ > 0.

其中r = 0, 1, 2, …，λ > 0。

A remarkable property is that the mean and variance are both equal to λ:

一个显著性质是均值与方差都等于λ：

E(X) = λ, Var(X) = λ

Poisson models are used for arrival times, telephone calls, defects in materials, and radioactive decays.

泊松模型用于到达时间、电话呼叫、材料缺陷和放射性衰变等。

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9. Geometric Distribution | 几何分布

The geometric distribution models the number of trials needed to obtain the first success in repeated independent trials with success probability p.

几何分布模拟在成功概率为p的重复独立试验中，获得首次成功所需的试验次数。

If X is the number of trials until the first success, then:

设X为直到首次成功所需的试验次数，则：

P(X = r) = (1−p)ʳ⁻¹ p

for r = 1, 2, 3, …

其中r = 1, 2, 3, …

The mean and variance are:

其均值和方差为：

E(X) = 1/p, Var(X) = (1−p)/p²

Example: If the probability of rolling a 6 is 1/6, the expected number of rolls to get the first 6 is 6.

例如：如果掷出6的概率是1/6，那么首次掷出6的期望试验次数为6。

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10. Applications of Probability Models | 概率模型的应用

Probability models are used in risk assessment, insurance pricing, quality control, and decision theory. For example, an

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