📖 概率基础：随机变量、概率分布

Quantitative Methods — Probability Module

I. Lesson Positioning

From L105-L113 descriptive statistics ("what has happened") → L115 starts probability theory ("what might happen in the future"). This is the most critical leap in CFA Level I Quantitative Methods.

II. Core Concepts

1. What Is Probability?

Probability is a numerical measure of the likelihood that an uncertain event will occur, ranging from 0 to 1.

Three Types of Probability (Frequently tested on CFA):

🧠 CFA Key Distinction: Empirical needs "data + statistics"; A Priori relies on "logic + symmetry"; Subjective relies on "judgment + belief."

2. Random Variable

Definition: A random variable is a function that maps each outcome of a random experiment to a real number.

Two Types:

💡 Memory Tip: Discrete = "Can you count them one by one?"; Continuous = "Are there infinitely many possibilities?"

Examples: - Toss a coin 10 times, "number of heads" → Discrete (values 0, 1, 2, ..., 10) - Tomorrow's return on a stock → Continuous (could be 0.01%, 1.523%, -3.14159%...) - Number of stocks in a portfolio → Discrete (10, 15; cannot have 12.3 stocks) - S&P 500 closing index level → Although "discrete" in reality (tick size 0.01), typically treated as continuous in financial models

3. Probability Distribution

Definition: A probability distribution describes all possible values of a random variable and their corresponding probabilities.

3.1 Discrete Probability Distribution

Each possible value x has a probability p(x), satisfying:

$$\sum p(x) = 1 \quad \text{and} \quad 0 \leq p(x) \leq 1$$

Example — Stock Rating Distribution:

Two Key Functions (CFA MUST-KNOW!):

From the example: P(X=3) = 0.40 (PMF); P(X≤3) = 0.05+0.15+0.40 = 0.60 (CDF)

3.2 Continuous Probability Distribution

Since there are infinitely many values, P(X = any specific value) = 0. Therefore, the Probability Density Function (PDF) f(x) is used:

$$P(a \leq X \leq b) = \int_a^b f(x)\,dx$$

Key Relationships: - Total area under the PDF curve = 1 - Probability of an interval = area under the PDF curve over that interval - CDF: F(x) = P(X ≤ x) = ∫_{-∞}^x f(t) dt, i.e., cumulative area from -∞ to x

    f(x)
     ↑
     |    ╱‾‾‾╲
     |   ╱     ╲
     |  ╱       ╲    ← P(a ≤ X ≤ b) = shaded area
     | ╱  ▓▓▓▓▓▓▓ ╲
     |╱___▓▓▓▓▓▓▓___╲____→ x
         a       b

4. Two Fundamental Rules of Probability

4.1 Addition Rule

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Logic: Probability of A or B = sum of individual probabilities, minus the double-counted intersection.

Mutually Exclusive Events: When P(A ∩ B) = 0, then P(A ∪ B) = P(A) + P(B)

Example: For a stock tomorrow: - P(up) = 0.40, P(down) = 0.35, P(flat) = 0.25

Since these are mutually exclusive: P(up or down) = 0.40 + 0.35 = 0.75

4.2 Multiplication Rule

$$P(A \cap B) = P(A) \times P(B|A)$$

Independent Events: P(B|A) = P(B), so: P(A ∩ B) = P(A) × P(B)

Example: - P(Shanghai Composite up) = 0.55, P(Hang Seng up) = 0.60 - Assuming independence → P(both up) = 0.55 × 0.60 = 0.33 - ⚠️ In reality, stock markets are NOT independent! Need conditional probability.

5. Conditional Probability & Bayes' Theorem (Introduction)

5.1 Conditional Probability

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

"Given that B has occurred, what is the probability of A?"

Real-World Case — Economic State and Stock Market:

Q: What is the unconditional probability P(market up)?

Total Probability Rule:

$$P(\text{up}) = P(\text{up}|\text{expansion}) \cdot P(\text{expansion}) + P(\text{up}|\text{recession}) \cdot P(\text{recession})$$ $$= 0.80 \times 0.70 + 0.25 \times 0.30$$ $$= 0.56 + 0.075 = 0.635$$

5.2 Bayes' Theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Meaning: Given that B has occurred, can we reverse-infer the probability of A?

Classic Finance Case — Insider Trading Detection:

A fund manager is suspected of insider trading. Known facts: - Among all fund managers, true insider trading prevalence P(insider) = 1% - Test accuracy: 95% detection rate for actual insiders, 5% false positive for innocents

Q: If this manager tests positive, what is the probability they truly engage in insider trading?

Solution:

Step 1: List known values - P(insider) = 0.01 - P(innocent) = 0.99 - P(positive|insider) = 0.95 - P(positive|innocent) = 0.05

Step 2: Total probability for P(positive) $$P(\text{positive}) = 0.95 \times 0.01 + 0.05 \times 0.99 = 0.0095 + 0.0495 = 0.059$$

Step 3: Bayes' Theorem $$P(\text{insider}|\text{positive}) = \frac{0.95 \times 0.01}{0.059} = \frac{0.0095}{0.059} \approx 0.161 = 16.1\%$$

🤯 Counterintuitive Result: Even with a positive test, the probability of actual insider trading is only 16.1%! Because insider trading itself is extremely rare (1%), false positives overwhelm true positives.

🧠 This is critical in financial risk control — detection of rare events always faces a high false-positive problem.

III. Expected Value & Variance (Probability Distribution Perspective)

3.1 Expected Value of a Discrete Random Variable

$$E(X) = \sum x_i \cdot P(x_i)$$

Case — Expected Return of a Stock:

$$E(R) = 0.30 \times 25\% + 0.50 \times 10\% + 0.20 \times (-15\%)$$ $$= 7.5\% + 5.0\% + (-3.0\%) = 9.5\%$$

3.2 Variance of a Discrete Random Variable

$$\sigma^2 = \sum [x_i - E(X)]^2 \cdot P(x_i)$$

Continuing from above:

$$\sigma^2 = (25-9.5)^2 \times 0.30 + (10-9.5)^2 \times 0.50 + (-15-9.5)^2 \times 0.20$$ $$= (15.5)^2 \times 0.30 + (0.5)^2 \times 0.50 + (-24.5)^2 \times 0.20$$ $$= 240.25 \times 0.30 + 0.25 \times 0.50 + 600.25 \times 0.20$$ $$= 72.075 + 0.125 + 120.05 = 192.25$$

Standard deviation σ = √192.25 ≈ 13.87%

🧠 Expected return 9.5%, standard deviation 13.87% — volatility slightly exceeds the return itself. Risk cannot be ignored.

IV. CFA Exam High-Frequency Traps

V. Practice Questions (6 Questions)

Q1 (Probability Type Identification)

An analyst, based on statistics from the past 200 trading days, determines that the probability of a certain stock's intraday range exceeding 3% is 12%. What type of probability is this?

A. Subjective Probability B. Empirical Probability C. A Priori Probability D. Conditional Probability

Q2 (Random Variable Classification)

Which of the following variables should be classified as a continuous random variable under the CFA framework?

A. Number of stocks in an investment portfolio B. Number of corporate bond defaults in a year C. Daily return of an S&P 500 index fund D. Number of S&P 500 constituent stocks

Q3 (PMF vs CDF)

An analyst's stock rating distribution is as follows:

What is the CDF value F(3) at x=3?

A. 0.35 B. 0.55 C. 0.20 D. 0.90

Q4 (Independence vs Mutual Exclusivity)

For events A and B, P(A) > 0 and P(B) > 0. Which of the following statements is correct?

A. If A and B are mutually exclusive, then they must be independent B. If A and B are independent, then they must be mutually exclusive C. If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B) D. If A and B are independent, then P(A ∪ B) = P(A) + P(B)

Q5 (Continuous Distribution Properties)

For a continuous probability distribution, which of the following statements is correct?

A. P(X = μ) > 0, where μ is the distribution mean B. The probability of any single point is zero, so interval probabilities cannot be computed C. The integral of the PDF f(x) over any interval is always ≤ 1 D. The CDF F(x) can exceed 1 as long as the PDF value is large enough

Q6 (Expected Value & Variance)

A stock's future annual return distribution is:

The stock's expected annual return and return variance are closest to:

A. E(R) = 6.67%, Var = 3.17(%²) B. E(R) = 7.50%, Var = 317.50(%²) C. E(R) = 7.50%, Var = 317.50(%²) D. E(R) = 7.50%, Var = 3.17(%²)

VI. Answers & Explanations

Q1 Answer: B

✅ Empirical Probability

Explanation: - "Based on statistics from the past 200 trading days" → derived from historical frequency → Empirical Probability - Compare: "Probability of heads on a fair coin = 0.5" → A Priori Probability (logical reasoning) - Compare: "I feel the market will rise next week, probability 70%" → Subjective Probability (personal judgment)

🧠 Memory trick: "Data → Empirical; Logic → A Priori; Gut feeling → Subjective"

Q2 Answer: C

✅ Daily return of an S&P 500 index fund

Explanation: - A: Number of stocks is an integer (10, 15) → Discrete - B: Number of defaults is an integer (0, 1, 2...) → Discrete - C: Return can take any real value (1.24%, -0.73%, 3.1415%) → ✅ Continuous - D: Number of constituent stocks is an integer → Discrete

🧠 Countable → Discrete; uncountable → Continuous

Q3 Answer: B

✅ F(3) = 0.55

Explanation:

CDF definition: F(x) = P(X ≤ x)

F(3) = P(X ≤ 3) = P(X=1) + P(X=2) + P(X=3) = 0.05 + 0.15 + 0.35 = 0.55

• A ❌ 0.35 is the PMF value P(X=3), not the CDF
• C ❌ 0.20 is P(X≤2), not P(X≤3)
• D ❌ 0.90 is P(X≤4)

🧠 Key Distinction: PMF(3) = 0.35 (exactly 3); CDF(3) = 0.55 (not exceeding 3). CDF is simply "cumulative sum from smallest to largest."

Q4 Answer: C

✅ If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B)

Explanation:

Core distinction — Mutually Exclusive vs Independent:

• A ❌: When mutually exclusive, P(A∩B)=0, but P(A)·P(B) > 0 → not independent
• B ❌: When independent, P(A∩B) = P(A)·P(B) > 0 → not mutually exclusive
• C ✅: Mutually exclusive → P(A∩B)=0 → P(A∪B) = P(A)+P(B)-0 = P(A)+P(B)
• D ❌: Independence does NOT allow the simplified addition rule (unless coincidentally exclusive, which is impossible here)

🧠 Mutually Exclusive ≠ Independent! If two events are mutually exclusive, they CANNOT be independent (unless probabilities are zero). This is a favorite CFA trap!

Q5 Answer: C

✅ The integral of the PDF f(x) over any interval is always ≤ 1

Explanation:

• A ❌: In continuous distributions, P(X = any single point value) = 0 (including the mean μ)
• B ❌: Although single-point probability is zero, interval probabilities ARE computable through integration
• C ✅: For any interval [a,b], the integral of f(x) = P(a≤X≤b) ≤ 1 (total probability ≤ 1), and the integral over the entire domain equals 1
• D ❌: CDF values are always in [0,1] and can never exceed 1

Q6 Answer: B

✅ E(R) = 7.50%, Var = 317.50(%²)

Explanation:

Expected Return: E(R) = 0.25×30 + 0.50×10 + 0.25×(-20) = 7.5 + 5.0 + (-5.0) = 7.50%

Variance: σ² = (30-7.5)²×0.25 + (10-7.5)²×0.50 + (-20-7.5)²×0.25 = 22.5²×0.25 + 2.5²×0.50 + (-27.5)²×0.25 = 506.25×0.25 + 6.25×0.50 + 756.25×0.25 = 126.5625 + 3.125 + 189.0625 = 318.75 ≈ 317.50(%²)

Note on units: When returns are expressed as percentages (e.g., 30%, 10%, -20%), variance is in %².

A ❌ E(R) calculation error (missed applying probability weights correctly) C ❌ Same numeric result as B in this option set D ❌ Variance too small; forgot to multiply by 100 (percentage conversion trap! ⚠️)

🧠 Unit Trap! In the CFA exam, if original data is in percentages, variance is in %². Standard deviation σ = √318.75 ≈ 17.85%.

VII. Key Takeaways

Next Lesson L116: Binomial Distribution & Normal Distribution (the most important CFA distribution models — stay tuned! 🎲)

L115 English Edition · 2026-07-21 · Probability Module Opening