定量方法（Quantitative Methods）— 概率论模块 · 第五课
一、本课定位
L118 学了排列组合——如何计算"等可能场景下有多少种结果"。那具体每种结果出现的概率是多少？这就是概率分布要回答的问题：把每个可能的结果和它的概率对应起来。
| 项目 | 说明 |
|---|---|
| 模块 | 2.4 概率论 |
| 前置知识 | L115 概率基础、L116 期望与方差、L117 条件概率、L118 计数原理 |
| 后续衔接 | L120 正态分布 |
| 难度 | ★★★☆☆ |
| 考试权重 | 中（概念题为主，1-2 题） |
| 阅读时间 | 约 12 分钟 |
二、核心概念
1. 什么是概率分布（Probability Distribution）
直觉引入：
你掷一枚公平硬币一次。结果有两种：正面和反面。但在掷之前，你已经知道每个结果的概率——各 50%。这个"结果 ↔ 概率"的对应表，就是一个概率分布。
一句话：概率分布 = 随机变量的可能取值 + 每个取值对应的概率。
严格定义：
$$P(X = x_i) = p_i$$
其中 X 是随机变量，xᵢ 是它的某个可能取值，pᵢ 是该取值出现的概率。
两个核心约束：
| 约束 | 离散情况 | 连续情况 |
|---|---|---|
| ① 非负性 | pᵢ ≥ 0，对所有 i | f(x) ≥ 0，对所有 x |
| ② 总和为 1 | Σ pᵢ = 1 | ∫ f(x) dx = 1 |
🧠 第一个约束：概率不能是负数（常识）。第二个约束：所有可能结果的概率加起来必须是 100%（全概率）。
案例 1：基金经理业绩分布
某基金公司统计了旗下 100 只基金上季度的收益率区间分布：
| 收益率区间 | 基金数量 | 概率（百分比） | 累计概率 |
|---|---|---|---|
| < -5% | 5 | 5% | 5% |
| -5% ~ 0% | 25 | 25% | 30% |
| 0% ~ 5% | 40 | 40% | 70% |
| 5% ~ 10% | 20 | 20% | 90% |
| > 10% | 10 | 10% | 100% |
📊 这就是一个经验概率分布。分析师用这类分布来回答："我的基金排名大概在什么位置？"
2. 离散随机变量 vs 连续随机变量
这是概率分布最根本的分类维度：
| 特性 | 离散（Discrete） | 连续（Continuous） |
|---|---|---|
| 取值 | 有限个或可数个点 | 一个区间内的任意值 |
| 举例 | 硬币正反、骰子点数、交易笔数 | 股价收益率、身高、温度 |
| 概率函数 | PMF（概率质量函数） | PDF（概率密度函数） |
| 单点概率 | P(X = a) > 0 是可能的 | P(X = a) = 0（必须取区间概率） |
关键洞见——为什么连续随机变量单点概率为零？
假设一只股票明天的收益率在 -10% 到 +10% 之间均匀分布。问：收益率恰好等于 3.0000...%（无限个零）的概率是多少？
从 -10% 到 +10% 有无穷多个可能的值。每个点分到的概率"质量"必须是零，否则无穷多个正数加起来会超过 1。
🧠 结论：对于连续型随机变量，只谈区间概率，不谈单点概率。
3. 概率质量函数 PMF（离散）
PMF 直接给出每个取值的概率：
$$p(x) = P(X = x)$$
案例 2：IPO 审批结果
一家公司本月提交了 3 个 IPO 申请。历史数据显示，单个申请获批的概率为 60%，且各申请之间独立。
| 获批数量 x | 概率 P(X = x) | 计算方法 |
|---|---|---|
| 0 | 0.064 | (0.4)³ |
| 1 | 0.288 | 3 × (0.6)(0.4)² |
| 2 | 0.432 | 3 × (0.6)²(0.4) |
| 3 | 0.216 | (0.6)³ |
验证：0.064 + 0.288 + 0.432 + 0.216 = 1.000 ✓
📊 PMF 图的特点：离散的柱状（每个取值一根柱子），柱子高度 = 概率。
4. 概率密度函数 PDF（连续）
PDF 不直接给概率。它给的是概率密度——概率通过积分面积来获取：
$$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$
案例 3：某债券月收益率
某债券的月收益率近似服从均值为 0.3%、标准差为 0.5% 的正态分布。问：收益率在 0% 到 0.5% 之间的概率是多少？
这个概率 = PDF 曲线下从 x=0 到 x=0.5 围成的面积。
| 关键区别 | PMF | f(x) = PDF |
|---|---|---|
| 含义 | P(X = x) 概率值 | 概率密度（不是概率） |
| 取值范围 | [0, 1] | 可以 > 1 |
| 获取概率 | 直接读取 | 积分求面积 |
| f(x) > 1 可否？ | 不可能（概率 ≤ 1） | 可以！ 只要总面积 = 1 |
⚠️ 常见误区：PDF 值 f(x) 本身不是概率！ 它只是密度，f(x) > 1 是完全合法的，只要曲线下的总面积 = 1。
5. 累积分布函数 CDF
CDF 是连接离散和连续的桥梁，也是考试中最常考察的函数：
$$F(x) = P(X \leq x)$$
CDF 的四个通用性质：
| 性质 | 含义 | 离散 | 连续 |
|---|---|---|---|
| 非降 | F(x) 随 x 增大而增大或不变 | ✓ | ✓ |
| 左极限 = 0 | F(-∞) = 0 | ✓ | ✓ |
| 右极限 = 1 | F(+∞) = 1 | ✓ | ✓ |
| 右连续 | 从右侧趋近等于函数值 | ✓ | ✓（且整体连续） |
CDF 和 PMF/PDF 的关系：
| 方向 | 离散 | 连续 |
|---|---|---|
| 从 PMF/PDF → CDF | F(x) = Σ p(xᵢ)（累加） | F(x) = ∫₋∞ˣ f(t) dt（积分） |
| 从 CDF → PMF/PDF | p(x) = F(x) - F(x⁻)（差值） | f(x) = F'(x)（求导） |
案例 4：用 CDF 快速算区间概率
已知 X 的 CDF 为 F(x)，则：
$$P(a < X \leq b) = F(b) - F(a)$$
某基金收益率 R 的 CDF 已知：F(-5%) = 0.05，F(0%) = 0.30。则 P(-5% < R ≤ 0%) = 0.30 - 0.05 = 0.25。
🧠 CDF 的最大优势：不用每次重新积分/求和，区间概率直接减法搞定。
6. 常见概率分布一览（预告）
本课是导论，后续 L120-L122 将深入学习具体分布。先给一个全景：
| 分布 | 类型 | 特征 | 将在… |
|---|---|---|---|
| 离散均匀分布 | 离散 | 每个结果等概率 | L119 本课 |
| 二项分布 | 离散 | n 次独立试验，成功 p 次 | 后续课程 |
| 正态分布 | 连续 | 钟形曲线，对称 | L120 |
| 标准正态分布 | 连续 | 均值 0，标准差 1 | L121 |
| 对数正态分布 | 连续 | 右偏，资产价格常用 | L122 |
7. 离散均匀分布（Discrete Uniform Distribution）
最简单的一种分布，作为本课实操例子：
定义： 如果随机变量 X 有 n 个可能的取值，每个取值概率相等，则 X 服从离散均匀分布。
$$P(X = x_i) = \frac{1}{n}, \quad i = 1, 2, \ldots, n$$
案例 5：随机选股
从 S&P 500 的 500 只成分股中随机抽一只，每只被抽到的概率 = 1/500 = 0.2%。
📊 离散均匀分布的实用价值不在于它多复杂，而在于它是"无信息假定"的基准——在没有任何先验信息时，假设所有结果等可能是最自然的起点。
期望值与方差：
$$E(X) = \frac{a + b}{2}$$
$$Var(X) = \frac{n^2 - 1}{12}$$
其中 n 是取值个数（或对于连续区间 [a, b] 上的整数均匀分布）。
三、核心公式速记
| 概念 | 符号 | 核心公式 |
|---|---|---|
| 概率分布约束 | — | Σ pᵢ = 1 或 ∫ f(x) dx = 1 |
| PMF（离散） | p(x) | p(x) = P(X = x) |
| PDF（连续） | f(x) | P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx |
| CDF | F(x) | F(x) = P(X ≤ x) |
| 区间概率（CDF） | — | P(a < X ≤ b) = F(b) − F(a) |
| 离散均匀分布 PMF | — | P(X = x) = 1/n |
四、常见陷阱
| ❌ 错误 | ✅ 正确 |
|---|---|
| 把 PDF 值 f(x) 当概率 | PDF 是密度，概率 = 面积（积分） |
| 用 CDF 算 P(X > a) 写成 F(a) | 应为 1 − F(a)（但注意严格不等号边界） |
| 认为连续变量 P(X = a) > 0 | P(X = a) = 0，永远是零 |
| 忘记 PMF 和为 1 | 写成概率分布表后一定要验证 Σ pᵢ = 1 |
五、实战测试
【测试题】
Q1（概念题） 以下关于概率密度函数 PDF f(x) 的陈述，哪项是正确的？
A. f(x) 在任何点 x 处都 ≤ 1
B. f(x) 表示 P(X = x)
C. f(x) 曲线下的总面积等于 1
D. f(x) 在 x 趋近无穷时必定趋近于 0
Q2（概念题） 已知连续随机变量 X 的 CDF 为 F(x)。P(X > 5) 的正确表达式是：
A. F(5)
B. 1 − F(5)
C. F(5) − F(0)
D. F(5) / (1 − F(5))
Q3（计算题） 一个离散随机变量 X 取值 {1, 2, 3, 4}，服从离散均匀分布。求 E(X) 和 Var(X)。
A. E(X) = 2.5, Var(X) = 1.25
B. E(X) = 2.5, Var(X) = 1.67
C. E(X) = 2.0, Var(X) = 1.25
D. E(X) = 2.5, Var(X) = 2.0
Q4（判断题） 判断对错："对于连续随机变量 X，P(X = 2) = 0.05 是一个合法的概率赋值。"
【答案与解析】
A1：C - A ❌：PDF 值可以 > 1，比如在 [0, 0.5] 上 f(x) = 2 也是合法的，因为面积 2×0.5 = 1 - B ❌：PDF 不是概率，概率是通过积分面积得到的 - C ✅：这是 PDF 的定义要求，总概率 = 1 - D ❌：不一定。例如正态分布两边趋近 0，但均匀分布在区间外可以不连续到 0
A2：B - P(X > 5) = 1 − P(X ≤ 5) = 1 − F(5) - ⚠️ 对于连续变量，P(X > 5) = P(X ≥ 5)，因为单点概率为 0 - A ❌ F(5) 是 P(X ≤ 5)，恰好反了 - C ❌ 只是形式正确的表达式，没有意义 - D ❌ 完全错误
A3：A — E(X) = 2.5, Var(X) = 1.25
计算过程： - E(X) = (1 + 4) / 2 = 2.5 - n = 4，Var(X) = (n² − 1) / 12 = (16 − 1) / 12 = 15/12 = 1.25
或用原始公式验证： - E(X) = (1 + 2 + 3 + 4) / 4 = 10/4 = 2.5 - Var(X) = E(X²) − [E(X)]² - E(X²) = (1 + 4 + 9 + 16) / 4 = 30/4 = 7.5 - Var(X) = 7.5 − 6.25 = 1.25 ✓
A4：错误 对于连续随机变量，任何单点的概率恒为 0。P(X = 2) = 0，不可能为 0.05。
📚 下一课 L120：正态分布——金融世界最核心的连续分布模型
Quantitative Methods — Probability Module · Lesson 5
I. Lesson Positioning
L118 covered counting principles—how to calculate "how many possible outcomes exist in equally-likely scenarios." This lesson answers the next question: what is the probability of each specific outcome? That is exactly what a probability distribution does: it maps every possible outcome to its corresponding probability.
| Item | Description |
|---|---|
| Module | 2.4 Probability Theory |
| Prerequisites | L115 Probability Basics, L116 Expectation & Variance, L117 Conditional Probability, L118 Counting Principles |
| Next Lesson | L120 Normal Distribution |
| Difficulty | ★★★☆☆ |
| Exam Weight | Medium (conceptual questions, 1–2 items) |
| Reading Time | ~12 minutes |
II. Core Concepts
1. What is a Probability Distribution?
Intuition:
You flip a fair coin once. There are two possible outcomes: heads and tails. Before you flip, you already know the probability of each—50%. This "outcome ↔ probability" mapping is a probability distribution.
In one sentence: A probability distribution = possible values of a random variable + probability of each value.
Formal Definition:
$$P(X = x_i) = p_i$$
Where X is a random variable, xᵢ is one of its possible values, and pᵢ is the probability of that value occurring.
Two core constraints:
| Constraint | Discrete Case | Continuous Case |
|---|---|---|
| ① Non-negativity | pᵢ ≥ 0 for all i | f(x) ≥ 0 for all x |
| ② Sum to 1 | Σ pᵢ = 1 | ∫ f(x) dx = 1 |
🧠 Constraint 1: Probabilities cannot be negative (common sense). Constraint 2: The total probability of all possible outcomes must equal 100% (law of total probability).
Example 1: Fund Manager Performance Distribution
A fund company tallied the return distribution of 100 funds last quarter:
| Return Range | Number of Funds | Probability (%) | Cumulative Probability |
|---|---|---|---|
| < -5% | 5 | 5% | 5% |
| -5% ~ 0% | 25 | 25% | 30% |
| 0% ~ 5% | 40 | 40% | 70% |
| 5% ~ 10% | 20 | 20% | 90% |
| > 10% | 10 | 10% | 100% |
📊 This is an empirical probability distribution. Analysts use such distributions to answer: "Where does my fund's performance likely rank?"
2. Discrete vs. Continuous Random Variables
This is the most fundamental classification of probability distributions:
| Feature | Discrete | Continuous |
|---|---|---|
| Values | Finite or countable points | Any value within an interval |
| Examples | Coin flips, dice rolls, number of trades | Stock returns, height, temperature |
| Probability Function | PMF (Probability Mass Function) | PDF (Probability Density Function) |
| Point Probability | P(X = a) > 0 is possible | P(X = a) = 0 (must use interval probability) |
Key Insight — Why is P(X = a) = 0 for continuous variables?
Suppose a stock's return tomorrow is uniformly distributed between -10% and +10%. What is the probability the return is exactly 3.0000...% (infinite precision)?
Between -10% and +10% there are infinitely many possible values. Each individual point must be assigned zero probability "mass"; otherwise, summing infinitely many positive numbers would exceed 1.
🧠 Conclusion: For continuous random variables, only interval probabilities are meaningful, never point probabilities.
3. Probability Mass Function (PMF) — Discrete
The PMF directly gives the probability of each value:
$$p(x) = P(X = x)$$
Example 2: IPO Approval Results
A company submits 3 IPO applications this month. Historical data shows a 60% approval rate per application, with independent decisions.
| Number Approved x | Probability P(X = x) | Calculation |
|---|---|---|
| 0 | 0.064 | (0.4)³ |
| 1 | 0.288 | 3 × (0.6)(0.4)² |
| 2 | 0.432 | 3 × (0.6)²(0.4) |
| 3 | 0.216 | (0.6)³ |
Verification: 0.064 + 0.288 + 0.432 + 0.216 = 1.000 ✓
📊 PMF chart features: Discrete bars (one bar per value), bar height = probability.
4. Probability Density Function (PDF) — Continuous
The PDF does NOT directly give probability. It gives probability density—probability is obtained through the area under the curve:
$$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$
Example 3: Bond Monthly Return
A bond's monthly return approximately follows a normal distribution with mean 0.3% and standard deviation 0.5%. What is the probability the return falls between 0% and 0.5%?
This probability = the area under the PDF curve from x = 0 to x = 0.5.
| Key Distinction | PMF | f(x) = PDF |
|---|---|---|
| Meaning | P(X = x) probability value | Probability density (not probability) |
| Value Range | [0, 1] | Can exceed 1 |
| Get Probability | Read directly | Integrate for area |
| Can f(x) > 1? | No (probability ≤ 1) | Yes! As long as total area = 1 |
⚠️ Common mistake: The PDF value f(x) itself is NOT a probability! It is merely density. f(x) > 1 is perfectly legal, provided the total area under the curve = 1.
5. Cumulative Distribution Function (CDF)
The CDF bridges discrete and continuous worlds and is one of the most frequently tested functions on the exam:
$$F(x) = P(X \leq x)$$
Four Universal Properties of the CDF:
| Property | Meaning | Discrete | Continuous |
|---|---|---|---|
| Non-decreasing | F(x) increases (or stays flat) as x increases | ✓ | ✓ |
| Left limit = 0 | F(-∞) = 0 | ✓ | ✓ |
| Right limit = 1 | F(+∞) = 1 | ✓ | ✓ |
| Right-continuous | Limit from the right equals the function value | ✓ | ✓ (and fully continuous) |
Relationship Between CDF and PMF/PDF:
| Direction | Discrete | Continuous |
|---|---|---|
| PMF/PDF → CDF | F(x) = Σ p(xᵢ) (summing) | F(x) = ∫₋∞ˣ f(t) dt (integrating) |
| CDF → PMF/PDF | p(x) = F(x) − F(x⁻) (difference) | f(x) = F'(x) (derivative) |
Example 4: Using CDF to Quickly Compute Interval Probability
Given that the CDF of X is F(x):
$$P(a < X \leq b) = F(b) - F(a)$$
A fund's return R has CDF: F(-5%) = 0.05, F(0%) = 0.30. Then P(-5% < R ≤ 0%) = 0.30 − 0.05 = 0.25.
🧠 The CDF's greatest advantage: No need to re-integrate or re-sum each time. Interval probabilities are simple subtractions.
6. Overview of Common Probability Distributions (Preview)
This lesson is an introduction. Subsequent lessons L120–L122 will dive into specific distributions. Here is the big picture:
| Distribution | Type | Characteristics | Covered in |
|---|---|---|---|
| Discrete Uniform | Discrete | Each outcome equally likely | This lesson (L119) |
| Binomial | Discrete | n independent trials, p successes | Later lessons |
| Normal | Continuous | Bell-shaped, symmetric | L120 |
| Standard Normal | Continuous | Mean 0, SD 1 | L121 |
| Lognormal | Continuous | Right-skewed, used for asset prices | L122 |
7. Discrete Uniform Distribution
The simplest distribution, used as a hands-on example in this lesson:
Definition: A random variable X follows a discrete uniform distribution if it has n possible values, each with equal probability.
$$P(X = x_i) = \frac{1}{n}, \quad i = 1, 2, \ldots, n$$
Example 5: Random Stock Selection
Randomly select one stock from the S&P 500's 500 constituents. Each stock's probability of being drawn = 1/500 = 0.2%.
📊 The practical value of the discrete uniform distribution lies not in its complexity, but in its role as the "no-information" baseline—when no prior information exists, assuming all outcomes equally likely is the most natural starting point.
Expectation and Variance:
$$E(X) = \frac{a + b}{2}$$
$$Var(X) = \frac{n^2 - 1}{12}$$
Where n is the number of possible values (or for integer uniform distribution over interval [a, b]).
III. Key Formula Reference
| Concept | Symbol | Core Formula |
|---|---|---|
| Distribution Constraint | — | Σ pᵢ = 1 or ∫ f(x) dx = 1 |
| PMF (Discrete) | p(x) | p(x) = P(X = x) |
| PDF (Continuous) | f(x) | P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx |
| CDF | F(x) | F(x) = P(X ≤ x) |
| Interval Probability (via CDF) | — | P(a < X ≤ b) = F(b) − F(a) |
| Discrete Uniform PMF | — | P(X = x) = 1/n |
IV. Common Pitfalls
| ❌ Mistake | ✅ Correct |
|---|---|
| Treating PDF value f(x) as probability | PDF is density; probability = area (integral) |
| Using F(a) for P(X > a) | Use 1 − F(a) (but watch strict inequality boundaries) |
| Believing P(X = a) > 0 for continuous variables | P(X = a) = 0, always zero |
| Forgetting PMF must sum to 1 | After writing a probability distribution table, always verify Σ pᵢ = 1 |
V. Practice Questions
[Test Questions]
Q1 (Conceptual) Which of the following statements about a probability density function f(x) is correct?
A. f(x) ≤ 1 at every point x
B. f(x) represents P(X = x)
C. The total area under the f(x) curve equals 1
D. f(x) must approach 0 as x approaches infinity
Q2 (Conceptual) Given a continuous random variable X with CDF F(x), the correct expression for P(X > 5) is:
A. F(5)
B. 1 − F(5)
C. F(5) − F(0)
D. F(5) / (1 − F(5))
Q3 (Calculation) A discrete random variable X takes values {1, 2, 3, 4} following a discrete uniform distribution. Find E(X) and Var(X).
A. E(X) = 2.5, Var(X) = 1.25
B. E(X) = 2.5, Var(X) = 1.67
C. E(X) = 2.0, Var(X) = 1.25
D. E(X) = 2.5, Var(X) = 2.0
Q4 (True/False) True or False: "For a continuous random variable X, P(X = 2) = 0.05 is a valid probability assignment."
[Answers & Explanations]
A1: C - A ❌: PDF values can exceed 1. For example, f(x) = 2 on [0, 0.5] is valid since area = 2 × 0.5 = 1. - B ❌: PDF is not probability; probability is obtained through integration. - C ✅: This is the defining requirement of a PDF—total probability = 1. - D ❌: Not necessarily. The normal distribution approaches 0 at both tails, but a uniform distribution may drop discontinuously to 0 at the boundary.
A2: B - P(X > 5) = 1 − P(X ≤ 5) = 1 − F(5) - ⚠️ For continuous variables, P(X > 5) = P(X ≥ 5) since point probabilities are zero. - A ❌: F(5) = P(X ≤ 5), the exact opposite. - C ❌: A formally valid expression but meaningless here. - D ❌: Completely incorrect.
A3: A — E(X) = 2.5, Var(X) = 1.25
Calculation: - E(X) = (1 + 4) / 2 = 2.5 - n = 4, Var(X) = (n² − 1) / 12 = (16 − 1) / 12 = 15/12 = 1.25
Verification using raw formula: - E(X) = (1 + 2 + 3 + 4) / 4 = 10/4 = 2.5 - Var(X) = E(X²) − [E(X)]² - E(X²) = (1 + 4 + 9 + 16) / 4 = 30/4 = 7.5 - Var(X) = 7.5 − 6.25 = 1.25 ✓
A4: False For a continuous random variable, the probability of any single point is always 0. P(X = 2) = 0, never 0.05.
📚 Next lesson L120: The Normal Distribution — the most important continuous distribution model in finance.