Lesson 1.3: The Normal Distribution (The "Bell Curve")

The "wow" moment comes from a discovery called the Central Limit Theorem. The theorem says, in simple terms: "If you take any random process (like flipping a coin, rolling a die, or stock market movements) and you add up a bunch of those random events, the final sum will always start to look like a Normal Distribution."

• The height of a person? It's the sum of thousands of tiny genetic and environmental factors. It follows a bell curve.
• The daily change in a stock? It's the sum of millions of different buy/sell decisions from different people. It also follows a bell curve.

This is why we use it in finance. The "randomness" we see in the market is really the sum of millions of tiny, independent, random decisions. Therefore, the "Bell Curve" is the natural "recipe" for modeling this randomness.

A normal distribution is defined by only two numbers, $\mu$ and $\sigma$. We write the "recipe" as: $N(\mu, \sigma^2)$.

For any normal distribution, no matter its $\mu$ or $\sigma$, the following is always true:

• 1 Standard Deviation (1σ): About 68% of all your data will fall within 1 "standard distance" of the mean (between $\mu - \sigma$ and $\mu + \sigma$).
• 2 Standard Deviations (2σ): About 95% of all your data will fall within 2 "standard distances" of the mean.
• 3 Standard Deviations (3σ): About 99.7% of all your data will fall within 3 "standard distances" of the mean.

If a stock's daily change follows $N(0, \$2)$, we now know that on ~68% of all trading days, the stock will finish somewhere between -$2 and +$2.