Lecture 2: Random Variables

What is a Random Variable?

A random variable X maps outcomes to numbers
Formally: X : \Omega \to \mathbb{R}
Example: X(\omega) = payoff of an option at expiry
It is a function, not a number

Welcome back. In Lecture 1 we built the scaffolding — sample spaces and events. Now we introduce the concept that makes probability useful in practice: the random variable.

A random variable, typically written as a capital letter such as X, is a function that maps every outcome in the sample space to a real number. The formal notation is X from Omega to the real line. For every possible outcome omega in Omega, X of omega is a specific real number.

Let me give you a concrete financial example. Suppose we are pricing a European call option. The underlying stock price at expiry is random — we do not know today what it will be. That stock price is our random phenomenon. The payoff of the call option — which is the maximum of zero and the stock price minus the strike — is a function of that stock price. So the payoff is a random variable: it maps each possible stock price outcome to a specific dollar amount.

Now here is the critical conceptual point I want you to remember: a random variable is a function, not a number. Before the experiment runs — before we observe which outcome actually occurs — X does not have a fixed value. It is a rule that says: if the outcome is this, then X equals that. Only after the outcome is observed does X take a concrete numerical value, which we call a realization of X.

The word “random” in the name refers to the fact that we do not know in advance which outcome will occur, and therefore we do not know in advance what value X will realize. But the function itself — the rule mapping outcomes to numbers — is completely deterministic. This distinction between the random variable as a function and its realized value is one of the most commonly confused points in probability, so I want to be very explicit about it here.

Discrete vs Continuous

Discrete: X takes countably many values
- Example: number of defaults in a portfolio
Continuous: X takes values in an interval
- Example: log-return of a stock over one day
Different tools for each type

Random variables come in two fundamental types, and the distinction between them determines which mathematical tools we use.

A discrete random variable takes a countable set of values. Countable means we can list the values: value number one, value number two, value number three, and so on — even if the list goes on forever. The values are typically integers or rationals. The number of defaults in a credit portfolio is a perfect example: it can be zero defaults, one default, two defaults, and so on. It cannot be two-and-a-half defaults. The mathematical object that describes a discrete random variable is the probability mass function, or PMF — a function that assigns a probability to each possible value X can take. These probabilities must sum to one across all possible values.

A continuous random variable takes values in an interval of the real line — or even the entire real line. There are no gaps, no discrete jumps. The log-return of a stock over a trading day is the canonical example in finance: it is modeled as a continuous random variable, because between any two values it can take, there is another value it could also take. For continuous random variables, the PMF does not work — the probability of any single exact value is zero, because there are uncountably many values. Instead we use the probability density function, or PDF, denoted f of x. The probability that X falls in a given interval is the integral of f over that interval.

There is a third category worth knowing: mixed random variables, which have both discrete and continuous components. A bond that defaults with probability p and otherwise pays a continuous return is a classic example. But for most of this course we will focus on pure discrete and pure continuous types.

Why does this distinction matter so much in practice? Because the entire apparatus of calculation changes. Expectations, variances, conditional distributions — all of these are computed with sums for discrete variables and integrals for continuous variables. Confusing the two leads to nonsensical results. The good news is that modern measure theory provides a unified framework that handles both simultaneously, but that level of generality is beyond our scope here.

Expected Value

The expectation averages over all outcomes
Discrete: \mathbb{E}[X] = \sum_x x \, P(X = x)
Continuous: \mathbb{E}[X] = \int_{-\infty}^{\infty} x \, f(x) \, dx
It is the probability-weighted average outcome

We now arrive at the single most important summary statistic in probability: the expected value, also called the expectation or the mean.

The expected value of X answers a very natural question: if we ran the random experiment a very large number of times, independently, and averaged all the realized values of X, what number would that average converge to? The expected value is that limiting average. It is the probability-weighted average of all possible values X can take.

For a discrete random variable, the formula is a sum: we multiply each possible value x by the probability that X equals x, and we add all of these up. Notice that values with high probability get more weight, and values with low probability get less weight. The expected value need not itself be a possible value of X — for a fair die, the expected value is three-and-a-half, which the die can never actually show.

For a continuous random variable, the sum becomes an integral: we integrate x times the probability density function f of x over the entire real line. The density f of x plays the same role as the probability P of X equals x, but for continuous variables it is a density, not a probability — you must integrate it over an interval to get a probability.

In both cases, the interpretation is the same: the expectation is the center of mass of the probability distribution. Imagine placing weights proportional to the probability at each point on the real line. The expected value is the point at which the whole thing perfectly balances.

In quantitative finance, expectations are absolutely everywhere. The risk-neutral pricing formula says that the fair price of a derivative today is the expected discounted payoff under a special probability measure called the risk-neutral measure. The expected return of a portfolio is the probability-weighted average return across all scenarios. The Value at Risk is defined in terms of quantiles, which are closely related to the distribution of a random variable, which is characterized by its expectation and higher moments.

So master the expected value — it is the foundation upon which all of financial mathematics is built.