Probability & Random Variables

You already know that a stock’s return next month is uncertain — that’s the whole reason volatility and the Sharpe ratio exist. But “uncertain” is a feeling, not a tool. To actually reason about returns — to say how likely a 5% drawdown is, or whether two stocks crash together — you need the formal grammar of uncertainty. That grammar is probability, and its central character is the random variable: a number whose value is decided by chance, like tomorrow’s return. This lesson builds that vocabulary from the ground up so every later statistics lesson has solid footing.

The building blocks: sample space, events, axioms

Analogy. Before you can talk odds at a casino, you agree on the board: a roulette wheel has 37 pockets, a die has 6 faces. Probability starts the same way — by writing down every outcome that could happen, then making honest bets about groups of them.

Definitions, in order.

• The sample space, written $\Omega$, is the set of all possible outcomes of a chance experiment. For one roll of a die, $\Omega = \{1,2,3,4,5,6\}$. For “did the market close up or down today?”, $\Omega = \{\text{up}, \text{down}\}$.
• An outcome is a single element of $\Omega$ — one specific way the world could turn out.
• An event is any subset of the sample space — a group of outcomes you care about. “The die shows an even number” is the event $\{2,4,6\}$. “The market rose at least 1%” is an event too. Events are the things we assign probabilities to.
• A probability $P(A)$ is a number attached to each event $A$, measuring how likely it is.

Those probabilities are not allowed to be assigned at random. They must obey Kolmogorov’s three axioms — the minimal rules that make probability coherent:

1. Non-negativity. $P(A) \ge 0$ for every event. You can’t be “−20% likely” to lose money.
2. Normalization. $P(\Omega) = 1$. Something in the sample space happens with certainty — the total probability is exactly one.
3. Additivity. If two events $A$ and $B$ are mutually exclusive (they share no outcomes, so they can’t both happen), then $P(A \text{ or } B) = P(A) + P(B)$. The chance of rolling a 1 or a 2 is $\tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}$.

Worked example. A fair die. The event “even” is $\{2,4,6\}$, three of the six equally likely outcomes, so $P(\text{even}) = \tfrac{3}{6} = 0.5$. The event “shows a 5” is one outcome, $P = \tfrac{1}{6} \approx 0.167$. Are “even” and “shows a 5” mutually exclusive? Yes — 5 is odd, so the two events share nothing. By additivity, $P(\text{even or 5}) = 0.5 + 0.167 = 0.667$, which is exactly $\tfrac{4}{6}$ — the four outcomes $\{2,4,5,6\}$. The arithmetic agrees with simply counting, which is the whole point: the axioms are just bookkeeping that never lets your bets contradict each other.

Two ways to read a probability. What does $P(\text{rain}) = 0.3$ even mean? There are two honest answers, and finance uses both:

• The frequentist reading: probability is the long-run frequency of an event over many independent repetitions. If you could rerun “this kind of day” a thousand times, it rains on about 300 of them. A coin’s $P(\text{heads}) = 0.5$ because over millions of flips, half land heads.
• The Bayesian reading: probability is a degree of belief — how confident a rational agent is, given the information they have. “There’s a 30% chance this merger closes” can’t be a long-run frequency (the merger happens once), so it’s a belief, updated as news arrives.

Both obey the same three axioms, which is why the math doesn’t care which interpretation you hold. We’ll lean frequentist when there’s repeatable data (returns) and Bayesian when there isn’t (one-off events).

Conditioning and independence: do two stocks move together?

Analogy. Learning that it’s cloudy changes your guess about rain — you’ve narrowed the world to “cloudy days” and re-asked the question inside that smaller world. Conditional probability is exactly that: re-computing a probability after you’re told some other event happened.

Definition. The conditional probability of $A$ given $B$, written $P(A \mid B)$, is the probability that $A$ happens once you know $B$ has happened:

You shrink the sample space down to $B$ (the new “certain” world), then ask what fraction of that world also has $A$. The denominator $P(B)$ renormalizes so the conditional probabilities still sum to one inside $B$.

Rearranging gives the multiplication rule — the probability that both happen:

Independence. Two events are independent when knowing one tells you nothing about the other: $P(A \mid B) = P(A)$. Conditioning didn’t move the needle. Plug that into the multiplication rule and independence collapses to a clean test:

Worked example — two stocks. Suppose on any given day stock $A$ falls with probability $P(A) = 0.4$ and stock $B$ falls with probability $P(B) = 0.5$. We also observe that they both fall on 30% of days, so $P(A \text{ and } B) = 0.3$. Are their crashes independent?

If the stocks were independent, they’d both fall on only 20% of days. We observe 30% — more than independence predicts. And $P(A \mid B) = 0.6$, well above the unconditional $P(A) = 0.4$: learning that $B$ fell raises your estimate that $A$ fell from 40% to 60%. These stocks are positively dependent — they tend to crash together. That co-movement is precisely what correlation measures and what diversification tries to escape; here we’ve shown the dependence directly, without ever computing a correlation.

Random variables: turning outcomes into numbers

Analogy. A sample space can be messy — “up/down”, “heads/tails”, a tangle of market states. A random variable is a translator that pins a number onto every outcome, so you can do arithmetic with chance instead of just listing scenarios.

Definition. A random variable (RV) is a function that maps each outcome in the sample space to a real number. We write RVs as capital letters: let $X$ be “the number of heads in two coin flips”, or let $R$ be “this stock’s return next month”. The outcome is random, so the number $X$ or $R$ is random too — hence the name.

RVs come in two flavors, and the distinction drives everything that follows:

• A discrete RV takes values from a countable list — you can enumerate them, often with gaps between. The number of heads in two flips is $\{0, 1, 2\}$. The number of defaults in a bond portfolio is $\{0, 1, 2, \dots, N\}$. A credit rating coded as $\{1,2,3,4,5\}$.
• A continuous RV can take any value in a range — an uncountable continuum with no gaps. A stock’s return can in principle be $7\%$, $7.0001\%$, or any real number near there. Time, price, and return are the classic continuous quantities in finance.

A return as a continuous RV. Model a monthly return $R$ as continuous — it can land anywhere on a smooth scale. This has a sharp, slightly unsettling consequence: the probability that $R$ equals exactly one specific value, say precisely $5.000000\%$, is zero. Not “small” — zero. There are infinitely many real numbers crowded around 5%, so no single one can carry positive probability without the total blowing past 1. For continuous RVs, probability only makes sense over intervals: $P(4\% \le R \le 6\%)$ is a sensible, positive number, while $P(R = 5\%)$ is not. Hold that thought — it’s the key to the next section.

Distributions: PMF, PDF, and CDF

A random variable isn’t fully described until you say how its probability is spread across its values. That spread is its distribution, and how you write it down depends on whether the RV is discrete or continuous.

Discrete: the probability mass function (PMF)

For a discrete RV, the probability mass function $p(x) = P(X = x)$ gives the probability of each individual value. It’s a list of point-masses: a height sitting on every value the RV can take. Because some value must occur, the masses sum to one: $\sum_x p(x) = 1$ (axiom 2 in disguise).

Worked example. Two fair coin flips, $X$ = number of heads. The four equally likely outcomes are TT, TH, HT, HH, so:

The masses sum to $0.25 + 0.50 + 0.25 = 1$. ✓ To get the probability of an event, you add up the relevant masses: $P(X \ge 1) = p(1) + p(2) = 0.75$.

Continuous: the probability density function (PDF)

A continuous RV can’t have a mass on each value — we just saw $P(R = 5\%) = 0$. Instead it has a probability density function $f(x)$: a smooth curve where probability is the area underneath, not the height. The density $f(x)$ is not a probability (it can even exceed 1); only the area over an interval is:

The total area under the whole curve is exactly 1 — normalization again. This is why a single point has zero probability: the area over a zero-width sliver is zero. The bell-shaped curves below are PDFs of returns; probability is the shaded area sitting between two return values.

Both: the cumulative distribution function (CDF)

The cumulative distribution function $F(x) = P(X \le x)$ answers one universal question — “what’s the probability of landing at or below $x$?” — and it works for discrete and continuous RVs alike. It starts at 0 on the far left, climbs to 1 on the far right, and never decreases (more ground covered can only add probability). For a continuous RV, $F(x)$ is the running area under the PDF up to $x$; for a discrete RV, it’s the running sum of masses, stepping up at each value.

Worked example. For the two-coin-flip $X$ above: $F(0) = 0.25$, $F(1) = 0.25 + 0.50 = 0.75$, and $F(2) = 1.00$. To get an interval probability you subtract: $P(X \le 2) - P(X \le 0) = 1.00 - 0.25 = 0.75 = P(X \ge 1)$, matching the PMF answer. For returns, the same trick gives any interval: $P(a \le R \le b) = F(b) - F(a)$. The CDF is the workhorse for “what’s the probability of a loss worse than $-10\%$?” — it’s just $F(-10\%)$.

Expectation: the probability-weighted average

Analogy. If you played a gamble a million times, what would your average payout settle at? That long-run average is the expectation (or expected value, or mean) of the random variable — the distribution’s center of gravity, the balance point of the PMF or PDF.

Definition. For a discrete RV, the expectation $E[X]$ is each value weighted by its probability mass and summed:

For a continuous RV the sum becomes an integral against the density, $E[X] = \int x\,f(x)\,dx$, but the idea is identical: every possible value, weighted by how likely it is. It is not a plain average of the values — a value that’s twice as likely pulls the mean twice as hard.

Worked example. A bet pays $10 with probability 0.2, $0 with probability 0.5, and costs you $4 (a $-4$ payout) with probability 0.3. The expected payout is each outcome times its probability, summed:

$E[X] = 2.0 + 0.0 - 1.2 = 0.8$, i.e. +$0.80 per play on average. Notice the expectation ($0.80) is a value the bet never actually pays — no single outcome is $0.80. That’s normal: the mean is a balance point, not a prediction of any one play. Applied to returns, $E[R]$ is the expected return you already met in the metrics course — now you can see it’s literally the probability-weighted average of every return the stock could deliver.

This is just the on-ramp. A later lesson gives expectation the full treatment — its algebra (linearity), its partner the variance (which is itself an expectation of squared deviations, and the squared sibling of the volatility you already know), and how both power portfolio math.

Putting it together

Probability is the grammar of uncertainty, and every later statistics lesson speaks it. You start with a sample space of outcomes, group them into events, and assign probabilities that obey three axioms — non-negativity, normalization, and additivity — which keep every bet between 0 and 1 and internally consistent. Conditioning ($P(A \mid B)$) re-asks a probability inside a smaller world, and independence is the special case where conditioning changes nothing — the lens through which we judged whether two stocks crash together. A random variable translates outcomes into numbers (discrete counts vs. continuous returns), its distribution spreads probability across those numbers (PMF for masses, PDF for area-under-density, CDF for “at or below”), and its expectation is the probability-weighted balance point — the formal definition of the expected return you’d been using all along.

Next up — descriptive statistics: now that returns are random variables with distributions, we’ll learn to summarize a pile of them with the mean, variance, and standard deviation (the formal home of the volatility you already use), then go further into skewness and kurtosis — the shape of the fat tails that decide how dangerous a return distribution really is.