Random variables — returns are random variables

The abstraction: random variables

A random variable is a function that assigns a number to each outcome of a random process. In plain terms: it's a machine that turns "whatever ends up happening" into a number. Two examples:

• Discrete: the number of heads in 10 coin flips. Possible values: 0, 1, ..., 10.
• Continuous: tomorrow's log return on SPY. (Log return — a return computed with logarithms; for small daily moves it is almost identical to the ordinary percent change.) Possible values: any real number (mostly small ones).

The capital convention is to use $X$ for the random variable itself and lowercase $x$ for a specific realization — the actual value that came out once the dust settled. So:

• $X$ = "tomorrow's SPY return" (the abstract concept, not yet known)
• $x = 0.0034$ (a specific realization, observed after tomorrow)

This distinction matters: a strategy is built around the distribution of $X$ (many possible futures), not the value of any single $x$.

Two flavors

Discrete

Takes one of countably many values — values you could list out one by one — each with its own probability:

$P(X = x_i) = p_i, \quad \sum_i p_i = 1$

In words: the probability that $X$ equals its i-th possible value $x_i$ is some number $p_i$. And when you add up all the $p_i$ (that's the $\sum$, a for-loop over every possible value), you must get 1 — because something has to happen.

Example: in a hypothetical "up day or down day" model:

• $P(X = \text{up}) = 0.53$
• $P(X = \text{down}) = 0.47$

That's a complete distribution.

Continuous

Takes any value in an interval. The probability of any single point is zero (this surprises people the first time). Instead, probability is described by a density function $f(x)$ — read "f of x." Picture it as a curve; its height at each value $x$ says how thickly probability is packed near that value:

$P(a \le X \le b) = \int_a^b f(x)\,dx$

In words: the probability that $X$ lands somewhere between $a$ and $b$ equals the area under the density curve between $a$ and $b$. The stretched-S symbol $\int$ is called an integral — it means "add up the area under the curve from $a$ to $b$." Read the whole thing aloud as: "the probability that X is between a and b equals the integral from a to b of f of x, d x."

Concrete example: if the density of tomorrow's SPY return is tallest near 0, the area between −1% and +1% — that is, $P(-0.01 \le X \le 0.01)$ — might be around 0.7. But the probability of any single exact value, like exactly 1.7%, is 0: a single point has no width, so it has no area.

For continuous random variables, we ask "what's the probability $X$ is between $a$ and $b$" — not "what's the probability $X$ equals exactly 1.7%."

Daily stock returns are usually modeled as continuous.

Why daily returns are well-modeled this way

Daily log-returns of liquid US equities — liquid meaning heavily traded, so easy to buy and sell — tend to look approximately like draws from a distribution centered slightly above zero, with some width. The exact distribution isn't perfectly normal. We'll come back to that in A1-03, where you'll meet fat tails — extreme moves that happen far more often than the bell curve predicts. But the abstraction itself is the foundational move. Treating returns as draws from a distribution we can describe is what lets us build:

• Risk metrics (variance of returns — variance is a measure of spread, formally defined in A1-04)
• Portfolio theory (mean and covariance of asset returns — covariance measures how two assets move together)
• Strategy evaluation (is the realized return distribution consistent with the hypothesis?)
• Option pricing (the entire Black-Scholes machinery)

Without random variables, none of this exists.

The cumulative distribution function (CDF)

A useful object for any random variable is the CDF:

$F(x) = P(X \le x)$

In words: F of x is the probability that the random variable comes out at or below the value $x$. As $x$ slides from very low to very high, $F(x)$ climbs from 0 up to 1.

For SPY daily returns: historically $F(-0.02) \approx 0.025$, meaning "about 2.5% of days have returns at or below −2%." The CDF lets you read off any "what fraction of days have returns below X?" question directly.

A related concept: quantiles — cut points of a distribution. The 5th percentile of daily returns is the value $x$ such that $F(x) = 0.05$: the point below which 5% of days fall. We'll use these constantly when we discuss Value-at-Risk — a standard "how much could I lose on a bad day" risk number — and tail risk, the risk of rare, extreme outcomes.

Explain it back

Without scrolling up, write 2–3 sentences explaining why we treat tomorrow's stock return as a random variable rather than just "an unknown number." Compare to the lesson when you're done.