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TL;DR

The bell curve distribution \( X \sim N(\mu, \sigma^2) \) with PDF \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)} \). Fully characterized by mean \( \mu \) and variance \( \sigma^2 \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Normal Distribution

The bell curve distribution \( X \sim N(\mu, \sigma^2) \) with PDF \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)} \). Fully characterized by mean \( \mu \) and variance \( \sigma^2 \).

Why it matters for interviews

The default distribution in quantitative finance. Black-Scholes assumes log-normal returns, portfolio theory uses multivariate normals, and most statistical tests assume normality. Understanding when it fails (fat tails) is equally important.

Definition and Mathematical Foundation

The bell curve distribution \( X \sim N(\mu, \sigma^2) \) with PDF \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)} \). Fully characterized by mean \( \mu \) and variance \( \sigma^2 \).

Application in Quantitative Finance

The default distribution in quantitative finance. Black-Scholes assumes log-normal returns, portfolio theory uses multivariate normals, and most statistical tests assume normality. Understanding when it fails (fat tails) is equally important.

Related Terms

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Frequently Asked Questions

Why is the normal distribution so prevalent in finance?
The CLT justifies normality for aggregated returns. It is analytically tractable (closed-form for sums, products with log-normal). Portfolio theory and Black-Scholes are built on it. However, empirical returns have fat tails.
What are the 68-95-99.7 rules?
Approximately 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean. In finance, extreme events occur more frequently than normal predicts (4+ sigma events).
What is the log-normal distribution and why use it for prices?
If \( \ln(X) \sim N(\mu, \sigma^2) \), then X is log-normal. Prices are modeled as log-normal because: (1) they cannot be negative, (2) percentage returns are additive, and (3) it arises naturally from GBM.