The Distribution Function: Compressing Chance
One monotone function on the real line carries the entire probabilistic content of a random variable. We prove the compression loses nothing — and meet a distribution that is neither coin-flip lumps nor a smooth density.
In this subchapter
4.2.1Definition and first reading
Fix a random variable X on (Ω, F, P) — the credential was earned in 4.1. Define, for each real t:
a non-decreasing function F: ℝ → [0,1]. Read it as a running total: stand at t on the number line, look left, and report how much probability mass has accumulated at or to the left of your position. The half-line questions of 4.1 were chosen precisely so that this function is well-defined on all of ℝ.
Coin flip (X = ±1 w.p. 1/2): a staircase with two jumps of height 1/2 — flat at 0 until t = −1, flat at 1/2 on [−1, 1), flat at 1 from t = 1 on. Uniform on [0,1]: F(t) = 0 for t < 0, F(t) = t on [0,1], F(t) = 1 beyond — the identity X(ω) = ω on Lebesgue space, at last with a graph. Waiting for a double-six: X = number of rolls, so F is a staircase with jumps (35/36)t−1(1/36) — geometric lumps marching right, thinning as they go.
4.2.2The three properties, and only these three
A function F: ℝ → ℝ is the distribution function of some random variable if and only if:
- F is non-decreasing: t₁ < t₂ implies F(t₁) ≤ F(t₂);
- F is right-continuous: F(t) = lims↓t F(s);
- limt→−∞ F(t) = 0 and limt→+∞ F(t) = 1.
Moreover, F determines the law of X completely (Theorem 3 below). The three properties are the whole fingerprint: nothing else about X survives the compression — or needs to.
That a CDF has the properties: monotonicity is inherited — if t₁ < t₂ then {X ≤ t₁} ⊆ {X ≤ t₂}, and probability respects containment (Chapter 2 axioms). The limits: {X ≤ −n} decreases to the impossible event (no outcome has X = −∞), so continuity from above (2.2 Theorem 3) sends F(−n) → 0; symmetrically {X ≤ n} ↑ Ω and continuity from below gives F(n) → 1. Right-continuity: {X ≤ t + 1/n} ↓ {X ≤ t}, continuity from above again. Each property is a continuity theorem of Chapter 2 wearing function clothes. The converse — every F with the three properties is some random variable's CDF — is a construction: take Ω = (0,1] with Lebesgue measure and let X(t) = inf{ x : F(x) ≥ t }, the "generalized inverse"; that F's three properties make X a random variable with CDF F is a verification worth doing once (5.4 uses the same device for simulating from any law). ∎
(a) F(t) = t on [0,1] but left-continuous at jumps elsewhere? Rejected by (2). Concretely: define F(t) = (1/2)·1[0,∞)(t) shifted so the jump at 0 is taken left-continuously, F(0) = 0 — a coin-flip-shaped staircase whose risers are on the wrong side; property (2) fails at t = 0, and correspondingly no random variable has this CDF (a variable with mass 1/2 at 0 must report F(0) = 1/2: the atom's mass is counted at its point). (b) F(t) = (1 + t/(1+|t|))/2 — monotone, right-continuous, limits 1/2 and 1: rejected by (3). (c) A decreasing segment anywhere: rejected by (1) and by common sense — mass cannot un-accumulate. The three properties are checkable in isolation, which is exactly why they are useful.
4.2.3Jumps are atoms
A point a ∈ ℝ is an atom of X (or of F) if P(X = a) > 0. Atoms are the probability lumps — the places where the running total jumps.
F has a jump at a of height F(a) − F(a−) := F(a) − lims↑a F(s) exactly when a is an atom, and the jump height is P(X = a). Between jumps, F can still creep (see §5) — jumps measure lumps, not all motion.
{X ≤ a} = {X ≤ a − 1/n} ∪ {a − 1/n < X ≤ a}, and as n → ∞ the events {X ≤ a − 1/n} increase to {X < a} (continuity from below, 2.2 Theorem 2). So F(a−) = P(X < a), and
by additivity of the difference (2.1). ∎
Given the staircase with jumps of 1/4 at t = 1, 1/4 at t = 2, and a linear creep from 1/2 to 1 over [2, 4]: read off — two atoms of mass 1/4; no other atoms; continuous spread of total mass 1/2 over [2,4] (uniform there); P(1 < X ≤ 3) = F(3) − F(1) = (1/2 + 3/16) − 1/4 = 7/16. The CDF is not a picture of the variable; it is a meter, and every interval probability is one subtraction.
Enumerate the rationals q₁, q₂, … and put mass 2−n at qn: a perfectly legal random variable (atoms only, no spread). Its CDF is a staircase with jumps at every rational — and it is still a staircase, with only countably many jumps, each of positive height. A monotone function can have at most countably many jumps (each jump hosts a rational in its gap), so: no CDF climbs entirely by jumping through uncountably many points. Continuous climbing — §5 — is a genuinely different mechanism, not just "lots of tiny atoms". A quant hearing "the distribution has a point mass at every rational and nowhere else" now knows exactly what the CDF looks like and what it doesn't.
4.2.4Uniqueness: the compression loses nothing
If X and Y are random variables (possibly on different probability spaces) with FX(t) = FY(t) for all t, then P(X ∈ B) = P(Y ∈ B) for every Borel set B: the two variables have the same law.
Let μ(B) = P(X ∈ B) and ν(B) = P(Y ∈ B): two probability measures on ℬ(ℝ) — measures, because measurability of X, Y (4.1 Theorem 1) makes every B-question an event. They agree on the π-system of half-lines (−∞, t] by hypothesis. The π–λ theorem of 1.4 — in its uniqueness corollary form: two finite measures agreeing on a π-system agree on the σ-algebra it generates — gives μ = ν on all of ℬ(ℝ). Note what the proof does not need: any relationship between the two underlying spaces. The law is a complete probabilistic passport; the CDF is its unique encoding by a single function. ∎
The theorem is about laws, not variables. Two variables with the same CDF can be wildly different as functions: on ([0,1], ℒ, λ), the identity X(ω) = ω and the reflection Y(ω) = 1 − ω have the same CDF (both uniform) but disagree at every single outcome. If your question involves joint behavior — "are X and Y ever simultaneously extreme?" — the CDFs of X and Y separately cannot answer it. That is the subject of 6.1 (independence) and beyond; the compression is lossless for one variable's law and blind to everything else.
4.2.5The Cantor distribution: continuous with no density
Take the Cantor staircase G from 3.3 — the continuous function that is constant on every removed middle third and climbs only on the null Cantor set. Theorem 1's three properties: monotone, continuous (hence right-continuous), limits 0 and 1 — certified. So G is a CDF; the random variable X = G−1(U) with U uniform (the generalized inverse of §2) has it. Its atoms: none (G never jumps — Theorem 2). Its density: none — if X had a density, the measure of the middle thirds would have to be positive, but P(X ∈ (1/3, 2/3)) = G(2/3) − G(1/3) = 1/2 − 1/2 = 0: mass 1/2 of probability sits on the closed thirds' complement, and a density integrating to 1/2 over a set of Lebesgue measure zero is impossible (the integral of a function over a null set is zero — 3.1). This distribution — continuous, atomless, and concentrated on the null Cantor set's worth of points — is the devil's staircase's revenge: an entire probability law whose mass is singular. Keep it in your zoo: it will be the canonical counterexample for conditioning (8.2) and for "continuous does not mean nice" throughout.
4.2.6What the compression keeps and loses
Assemble the ledger. The CDF determines: every interval probability (one subtraction), every Borel probability (π–λ, Theorem 3), atoms and their masses (Theorem 2), continuity or discreteness of the law, tails. It discards: everything about the underlying experiment — the space, the σ-algebra, the outcome-by-outcome behavior of X, and all joint structure with any other variable. This is a feature, not a bug: two questions with the same CDF are the same question as far as one-variable probabilities go, which is why quants can move freely between "a Gaussian variable on my P&L space" and "a Gaussian variable built from uniforms" without ever specifying the space. The remainder of the course exploits this: expectation (Chapter 5) can be computed from the law alone; convergence in distribution (Chapter 7) is a statement about CDFs and nothing else.
Historical simulation estimates a CDF from data; the empirical staircase is itself a CDF (jumps of 1/n at each observation — check the three properties). VaR at level α is read as the generalized inverse: VaRα = inf{ t : F(t) ≥ α } — the same device as the converse construction of Theorem 1. Expected shortfall integrates the jump heights. None of these operations needs the underlying space, and Theorem 3 guarantees two desks using the same fitted F are pricing the same law regardless of what their sample spaces look like. The formal license for swapping spaces at will arrives in 6.1.
One monotone, right-continuous function on ℝ is the complete story of a random variable's law — nothing more is needed, and nothing less suffices.
Random variables now have passports. But the definition in 4.1 has been hiding a second story: "X is measurable with respect to F" also reads "the information in F suffices to answer every question about X". The next subchapter makes information first-class — σ(X), the smallest σ-algebra that sees X, and the Doob–Dynkin lemma that says functions of X are exactly the variables measurable with respect to σ(X). That is the grammar on which conditioning (Chapter 8) is built.