This lesson is a quick review of the absolute fundamentals. If you've taken a basic statistics course, this will be familiar. If not, this is the most important foundation you will build.
Lesson 1.1: Key Concepts - Mean and Variance
Welcome to Module 1! Before we can model the 'random walk' of a stock price, we need to understand the two most important numbers that describe any random process: its 'center' and its 'spread'.
The 'Center' of a Random Event: The Mean (μ)
The Mean, or Expected Value, is the "long-run average" of a random process. We use the Greek letter (mu) to represent it.
The "Balancing Point" Analogy
Imagine a seesaw. If you place weights at different positions, the Mean is the single point where you could place the fulcrum to make the seesaw perfectly balance. It's the "center of gravity" of all the possible outcomes.
For a stock's daily return, the mean () is the "drift" or "trend" we expect it to have over the long run (e.g., +0.03% per day).
The "Spread" of a Random Event: The Variance (σ²)
Knowing the "center" isn't enough. Consider two stocks:
- Stock A (a utility company): Its returns are always very close to its average of 8% per year.
- Stock B (a biotech startup): Its returns are wild. It might be up 50% or down 40%, but it also averages 8% per year.
They have the same Mean, but Stock B is much "riskier." We need a number to measure this "risk," "spread," or "messiness." That number is the Variance.
Variance () measures the average *squared* distance of each outcome from the mean.
We square the distances so that big negative values and big positive values both count as "far from the center."
What's Next? (The 'Hook')
The Variance () is mathematically pure, but its units are weird (e.g., "percent-squared"). This makes it hard to talk about.
In our next lesson, we will introduce a simple "fix" for this problem: the Standard Deviation (). It is the single most important number for measuring risk in all of finance.