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Quantitative Methods — Probability Module · Lesson 6

1. Lesson Positioning

L119 covered the fundamental framework of probability distributions: what PMF, PDF, and CDF each do. This lesson dives into the most important continuous distribution in finance—the Normal Distribution. From asset return modeling to risk management VaR, from hypothesis testing to Monte Carlo simulation, the normal distribution is everywhere.

2. Core Concepts

2.1 What Is the Normal Distribution?

Intuition:

Measure the heights of 1,000 adult males. Most fall around 170 cm, a few are notably shorter or taller, and very few are extremely short or tall. Plotting the frequencies as a histogram yields a beautiful "bell shape"—peaked in the middle, tapering on both sides, perfectly symmetric. That is the shape of the normal distribution.

In a nutshell: The normal distribution is a bell-shaped symmetric continuous distribution fully defined by its mean μ and standard deviation σ.

Formal Definition:

The probability density function (PDF) of the normal distribution is:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \cdot e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Notation: $$X \sim N(\mu, \sigma^2)$$

Where: - μ (mu) = population mean (determines the center of the bell) - σ (sigma) = population standard deviation (determines the spread of the bell) - σ² = population variance

🧠 You do not need to memorize the PDF formula itself, but you must understand: μ and σ are the only two parameters needed to completely define a normal distribution.

2.2 Six Key Properties of the Normal Distribution

Property ② Explained—Mean = Median = Mode:

📊 The normal distribution's "three-in-one" property at the center is what makes many financial models tractable. If a distribution violates this, you cannot blindly apply the normal assumption.

Example 1: Normal vs. Non-Normal Identification

An analyst compiles the monthly return distributions of two funds:

Fund B's mean has been "pulled up"—a handful of high-return months drag the average higher, even though most months yield no more than 1%. This means you cannot model Fund B with a normal distribution.

2.3 The Empirical Rule: 68–95–99.7

This is the most practical rule about the normal distribution and a guaranteed exam topic:

Memory Aid:

μ ± 1σ → 68%   (think: "one sigma covers 68")
μ ± 2σ → 95%   (think: "two sigmas cover 95")
μ ± 3σ → 99.7% (think: "three sigmas cover almost all")

Example 2: Daily Stock Market Returns

Suppose S&P 500 daily returns are approximately normally distributed as N(0.04%, 1.2%²): - μ = 0.04% (average daily gain) - σ = 1.2% (daily volatility)

📊 If you observe two drops exceeding 3.56% within five trading days, that is clearly not explained by the normal distribution—either the model assumption is wrong, or a structural event has occurred.

Example 3: Quick Calculation Using 68–95–99.7

A normally distributed random variable X ~ N(100, 25). That is, μ = 100, σ = 5. Find: P(90 < X < 110) = ?

• 90 = 100 − 2×5 = μ − 2σ
• 110 = 100 + 2×5 = μ + 2σ
• Therefore P(90 < X < 110) ≈ 95%

No table lookup needed, no calculator required—the answer is immediate.

2.4 Linear Transformations Preserve Normality

A critical property: any linear combination of independent normal random variables is itself normally distributed.

If X ~ N(μₓ, σₓ²) and Y ~ N(μᵧ, σᵧ²), and X and Y are independent:

⚠️ Critical Trap: The variance of X − Y is also σₓ² + σᵧ² (addition! not subtraction!) Variances always represent "volatility stacking"—they add regardless of whether you are adding or subtracting the variables.

Example 4: Portfolio Return Distribution

Suppose you hold two independent stocks: - Stock A: monthly return ~ N(2%, 4%²) - Stock B: monthly return ~ N(1%, 3%²)

For an equally-weighted portfolio (50% each), the monthly return distribution is:

$$R_p = 0.5R_A + 0.5R_B$$

$$R_p \sim N(0.5 \times 2\% + 0.5 \times 1\%, \ 0.5^2 \times 16 + 0.5^2 \times 9)$$

$$R_p \sim N(1.5\%, \ 4 + 2.25 = 6.25\%^2)$$

Portfolio variance = 6.25, portfolio standard deviation = 2.5%.

📊 The portfolio mean is a simple weighted average (1.5%), but the portfolio standard deviation (2.5%) is lower than the weighted-average standard deviation (0.5×4% + 0.5×3% = 3.5%), illustrating the diversification benefit of risk reduction.

2.5 The Normal Distribution and Financial Applications

The normal distribution is pervasive in finance for two main reasons:

Specific Financial Applications:

⚠️ Reality Check: Real financial data often exhibit fat tails, meaning extreme events occur more frequently than the normal distribution predicts. The 2008 financial crisis powerfully demonstrated the limitations of the normality assumption. Nevertheless, the normal distribution remains the logical starting point for all financial models.

2.6 Normal vs. Other Common Distributions

L122 will cover the lognormal distribution in detail. Key insight: Prices → lognormal, Returns → normal.

3. Key Formula Cheat Sheet

4. Common Pitfalls

5. Practice Test

[Questions]

Q1 (Concept) Which of the following statements about the normal distribution N(μ, σ²) is incorrect?

A. The distribution is perfectly symmetric about the mean μ
B. The mode of the distribution equals μ
C. The skewness of the distribution is 0
D. The density function f(x) intersects the horizontal axis at x = μ ± 2σ

Q2 (Calculation) A stock's weekly return follows a normal distribution N(0.3%, 1.6%²). Using the 68–95–99.7 rule, the probability that the weekly return falls between −2.9% and 3.5% is closest to:

A. 68%
B. 90%
C. 95%
D. 99.7%

Q3 (Calculation) Given X ~ N(50, 64) and Y ~ N(30, 36), with X and Y independent, find the variance of 2X − Y:

A. 128
B. 156
C. 292
D. 292 × √2

Q4 (Concept) An analyst finds that for a fund's monthly return distribution, the mean = 1.2% and the median = 0.8%. This best indicates:

A. The fund's returns follow a normal distribution
B. The fund's return distribution is negatively skewed (left-skewed)
C. The fund's return distribution is positively skewed (right-skewed)
D. The fund's return volatility is negative

[Answers & Explanations]

A1: D

• A ✅ Normal distribution is perfectly symmetric—correct
• B ✅ Mode = μ (bell peaks at the center)—correct
• C ✅ Symmetry ⇒ skewness = 0—correct
• D ❌ The normal curve is asymptotic to the horizontal axis—it never actually touches it! This is the asymptotic tail property

A2: C — 95%

Calculation: - μ = 0.3%, σ = 1.6% - Lower bound: −2.9% = 0.3% − 2 × 1.6% = μ − 2σ - Upper bound: 3.5% = 0.3% + 2 × 1.6% = μ + 2σ - μ ± 2σ covers approximately 95%

🧠 Key technique: When you see interval boundaries, first express them as μ ± kσ, then match to 68/95/99.7.

A3: C — 292

Calculation: - Var(2X) = 2² × Var(X) = 4 × 64 = 256 - Var(−Y) = (−1)² × Var(Y) = 1 × 36 = 36 - Var(2X − Y) = Var(2X) + Var(−Y) = 256 + 36 = 292

⚠️ Note: Variance of 2X − Y = 4σₓ² + σᵧ². It is addition, never subtraction!

A4: C — Positively skewed (right-skewed)

• Mean (1.2%) > Median (0.8%) indicates a longer tail on the right side of the distribution
• This means a few high-return months have pulled the mean upward
• Positive (right) skew ⇒ the direction of skewness matches the direction of the tail
• A ❌ Normal distribution has mean = median
• B ❌ Negative skew would have mean < median
• D ❌ Volatility cannot be negative

📚 Next Lesson L121: Standard Normal Distribution & z-Score—how to use z-tables to compute exact probabilities for any interval