Financial returns are inherently uncertain. We can’t know for sure what a stock’s return will be next month, but we can model the likelihood of different outcomes. Think of a probability distribution as a tool that maps every possible outcome—like next month’s stock return—to its specific probability of occurring.

Descriptive statistics like the mean and standard deviation tell us the story of what has happened. Probability distributions, by contrast, provide a powerful framework for modeling what could happen, assigning a likelihood to every possible future outcome.

3.1. The Normal Distribution (The “Bell Curve”)

The normal distribution is the most common probability distribution used in finance. It is characterized by its symmetrical, “bell-shaped” curve and is defined by just two parameters:

• The mean (μ), which sets the center of the curve.
• The standard deviation (σ), which determines how wide or narrow the curve is. A larger standard deviation results in a flatter, wider curve, indicating greater dispersion (risk).

A practical way to interpret a normal distribution is through the Empirical Rule:

• Approximately 68% of all outcomes fall within 1 standard deviation of the mean.
• Approximately 95% of all outcomes fall within 2 standard deviations of the mean.
• Nearly 100% of all outcomes fall within 3 standard deviations of the mean.

3.2. Beyond the Bell Curve: “Fat Tails” in Finance

While the normal distribution is a useful starting point, it often fails to capture a critical feature of financial markets: the tendency for extreme events to occur more frequently than predicted. This is where other distributions, like the Student’s t-distribution, become valuable.

The single most important difference is that the Student’s t-distribution has “fatter tails” than the normal distribution. In a real-world context, “fat tails” mean that extreme outcomes—such as major market crashes or powerful rallies—are more likely to happen than a normal distribution would suggest. Because it better accounts for the probability of these rare but impactful events, the t-distribution is often considered a more realistic model for financial returns.

Now, let’s see how these individual statistical concepts can be combined into a simple but powerful financial model.