Random Variables & Probability Distributions

Now we turn an uncertain future outcome into something we can actually work with. Tomorrow's return is unknown, but it is not shapeless. It has a set of possible values and a likelihood for each. That idea has a name, the random variable, and the full description of its likelihoods is called a probability distribution. These two ideas are the foundation for everything else in this unit.

What a random variable is

A random variable is a variable whose value is a numerical outcome of a random process. Before the outcome is known, it represents all the values that could occur and how likely each is. The classic simple example is a die roll: the value is a random variable that could be 1, 2, 3, 4, 5, or 6, each with probability one sixth. In finance, the return a stock will deliver tomorrow is a random variable. You do not know its value yet, but you can describe the possibilities. So is the number of up days next month, or the price of an asset at some future date.

Discrete versus continuous random variables

• A discrete random variable takes separate, countable values, like a die roll (1 to 6) or the number of up days in a month.
• A continuous random variable can take any value in a range, like a stock's return, which could be 1.37 percent or negative 0.42 percent or anything between.

In finance, returns and prices are usually treated as continuous, while counts of events are discrete. This is the same discrete-versus-continuous split from the variables lesson, now applied to uncertain outcomes.

The variable versus a realization

Here is a subtle distinction that prevents a lot of confusion. The random variable is the concept, the full set of possible outcomes with their probabilities, before anything happens. A realization is one specific value that actually occurred. Tomorrow's return is a random variable. The return that yesterday actually delivered, now a fixed known number, is a realization of that random variable. When you look at a history of past returns, you are looking at a series of realizations of the underlying random process.

A random variable is the set of all outcomes that could happen. A realization is the single one that did.

What a probability distribution describes

A probability distribution spreads a total probability of one across all the possible values of a random variable, showing where the likelihood is concentrated and where it thins out. Because something must happen, the probabilities across all outcomes always add up to exactly one. The distribution captures not just the average outcome but the whole shape of what could occur, including how spread out the possibilities are and how likely the extremes are. Learning to think in whole distributions rather than single numbers is one of the big mental shifts in this unit.

Discrete and continuous distributions

For a discrete random variable like a die roll, the distribution simply lists the probability of each value, and those probabilities sum to one. For a continuous random variable like a return, there are infinitely many possible values and the probability of landing on any exact value is essentially zero. Instead, a smooth curve describes the relative likelihood across the range, and probability is measured as the area under that curve over an interval. The probability that a return falls between two values is the area beneath the curve between them, and the total area under the whole curve equals one. Do not worry about the exact math yet. The key picture is that probability for continuous data is an area under a curve.

Distributions in finance

In finance, we model returns and prices with probability distributions so we can reason about likely outcomes and quantify risk. The choice of distribution matters enormously. The most common assumption, the normal distribution, gets its own lesson later, after you have met the mean and standard deviation that describe it. For now, hold onto the big idea: an assumed distribution is a model of reality, not reality itself, and later you will see why choosing one that ignores extreme events is one of the costliest mistakes in quantitative finance.

You can now describe an uncertain outcome with a distribution. The next several lessons build the numbers that summarize a distribution, starting with the simplest question of all: where is its center?