Normal Distribution & LLN

The Bell Curve

The normal distribution (or Gaussian distribution) is the single most important probability distribution in quantitative finance. Its symmetric, bell-shaped density curve arises naturally whenever a quantity is influenced by many small, independent effects. Daily stock returns, measurement errors, and portfolio P&L fluctuations all tend to cluster around a central value and thin out in the tails — precisely the shape the normal distribution captures.

The PDF Formula

A continuous random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$ if its probability density function is

$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}}, \quad x \in (-\infty, \infty).$

The two parameters fully characterize the distribution:

• $\mu$ (mean) — the center of the bell, where the peak occurs. It determines the location of the distribution on the number line.
• $\sigma$ (standard deviation) — controls the spread. A larger $\sigma$ produces a wider, flatter bell; a smaller $\sigma$ yields a taller, narrower one. The variance is $\sigma^2$.

We write $X \sim N(\mu, \sigma^2)$ to denote that $X$ is normally distributed.

The 68-95-99.7 Rule

For any normal distribution, the probability of falling within one, two, or three standard deviations of the mean is fixed:

$P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.6827$

$P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.9545$

$P(\mu - 3\sigma \le X \le \mu + 3\sigma) \approx 0.9973$

This rule gives quick mental estimates. If someone tells you a hedge fund's daily P&L has mean $0 and standard deviation $1M, you immediately know that a $3M loss day should occur roughly $0.13\%$ of the time — about once every 750 trading days.

Z-Scores and Standardization

Any normal variable can be converted to a standard normal $Z \sim N(0,1)$ via the transformation

$Z = \frac{X - \mu}{\sigma}.$

This $Z$-score tells you how many standard deviations $X$ sits above or below the mean. Standardization lets you use a single standard normal table (or function $\Phi(z) = P(Z \le z)$) for every normal problem.

Key standard normal values worth memorizing:

| $z$ | $\Phi(z)$ | |-----|-----------| | 1.00 | 0.8413 | | 1.65 | 0.9505 | | 1.96 | 0.9750 | | 2.33 | 0.9901 |

The Law of Large Numbers

Before asking what distribution the sample mean follows, we should ask: does it converge at all? The Law of Large Numbers (LLN) answers yes.

If $X_1, X_2, \ldots$ are iid with finite mean $\mu$, then the sample mean $\bar{X}_n = (X_1 + \cdots + X_n)/n$ converges to $\mu$ as $n \to \infty$. The weak LLN says convergence is in probability; the strong LLN says it is almost sure.

In finance, LLN is why backtests work: if you evaluate a strategy over enough independent trades, the average P&L converges to the true expected P&L. The catch is "enough" — for heavy-tailed distributions (which real returns exhibit), convergence can be slow.

Why the Normal Distribution Appears Everywhere

The Central Limit Theorem (covered in the next lesson) goes further: not only does $\bar{X}_n$ converge to $\mu$ (LLN), but its distribution is approximately normal for large $n$. Because many financial quantities — a portfolio return, for instance — are sums of many small, roughly independent contributions, the normal approximation is remarkably useful.

Applications in Finance

• Modeling daily returns. Log-returns of liquid assets are often approximated as $r_t \sim N(\mu, \sigma^2)$, though real returns exhibit heavier tails.
• Value at Risk (VaR). Under a normal assumption, the $\alpha$-level VaR is $\text{VaR}_\alpha = \mu + z_\alpha \, \sigma$, where $z_\alpha$ is the corresponding quantile.
• Option pricing. The Black-Scholes model assumes log-returns are normally distributed, leading to the lognormal price process.

Worked Example

A stock's daily log-return is modeled as $R \sim N(0.0005, \, 0.02^2)$. What is the probability that the return on a given day exceeds $3\%$?

Step 1 — Standardize. Compute the $Z$-score for $R = 0.03$:

$Z = \frac{0.03 - 0.0005}{0.02} = \frac{0.0295}{0.02} = 1.475.$

Step 2 — Look up the probability. From the standard normal table, $\Phi(1.475) \approx 0.9299$.

Step 3 — Complement. We want $P(R > 0.03)$:

$P(R > 0.03) = 1 - \Phi(1.475) \approx 1 - 0.9299 = 0.0701.$

There is roughly a 7% chance the stock gains more than 3% on any given day.

Interview tip: Many quant interviews test whether you can quickly standardize and use the normal CDF. Memorize $\Phi(1) \approx 0.84$, $\Phi(1.65) \approx 0.95$, and $\Phi(2) \approx 0.975$ — these three values cover most back-of-the-envelope calculations.