Cumulative distribution functions: Visual walkthrough

Before any symbols, let us agree on the words.

Step 1 — Draw the number line and the question

WHAT. We put every possible outcome of $X$ on a horizontal number line, then pick a marker value we will call $x$ and ask: how much probability lies to the left of (or exactly on) that marker?

WHY. Every idea about "cumulative" needs a direction to accumulate along. The number line gives us that direction: we always sweep left → right, from "smallest possible outcome" toward "largest." The marker $x$ is the finish line of our sweep.

PICTURE. In the figure, the amber marker is $x$ . Everything shaded cyan to its left is what we are counting. The symbol $\leq$ (read "less than or equal to") means we include the marker itself, not just what is strictly before it.

Figure — Cumulative distribution functions

$the running-total function F_{X} (the finish line x) = "how much probability" P (outcome lands at or before x X \leq x)$

Read the whole line as: " $F_{X}$ evaluated at $x$ equals the probability that the unknown number $X$ turns out to be at or below $x$ ."

Step 2 — The discrete case: stacking blocks of probability

WHAT. Suppose $X$ can only be certain separate values (like a die: $1, 2, 3, 4, 5, 6$ ). Each value $x_{i}$ carries a block of probability written $p_{X} (x_{i}) = P (X = x_{i})$ — literally "the chance of landing exactly on $x_{i}$ ." To get the running total up to $x$ , we add every block whose value is $\leq x$ .

WHY. Separate outcomes cannot happen at the same time — rolling a $2$ and rolling a $3$ are mutually exclusive (they exclude each other). A basic law of probability says the chance of "this OR that" for exclusive events is just the sum of their chances. So accumulating = adding blocks.

PICTURE. Each die value is a block of height $\frac{1}{6}$ . As the amber marker $x$ slides right, it swallows one more block at each integer, and the running total jumps up by $\frac{1}{6}$ .

$F_{X} (x) = \sum_{x_{i} \leq x} height of one block p_{X} (x_{i})$

The symbol $\sum$ (a capital Greek "S," for Sum) just says "add up all the terms that match the condition written underneath." Here the condition is $x_{i} \leq x$ : only blocks to the left of the marker are added.

Step 3 — Why the discrete CDF is a staircase

WHAT. We now draw the whole function $F_{X} (x)$ , not just one value. Between two die values nothing changes (no new block is swallowed), so the graph is flat. At each die value it jumps by that value's block height.

WHY. This visual — flat then jump, flat then jump — is exactly the parent note's "staircase." The height of each step equals the point probability $p_{X} (x_{i})$ , and this gives us a way to recover point probabilities from a CDF later: the jump size at a point is its probability.

PICTURE. Six equal risers of height $\frac{1}{6}$ . The dots show the value is included at the jump (right-continuous), and open circles show it is not yet reached just before.

$P (X = x) = top of the step F_{X} (x) - value just before the step F_{X} (x^{-})$

Here $x^{-}$ means " $x$ from the left, an infinitesimal hair below." Their difference is the height of the riser — the jump.

Step 4 — The continuous case: no blocks, only a smear

WHAT. Now let $X$ be able to land anywhere on an interval (a bullet's exact distance). There are infinitely many possible values, so no single value can carry a block — its probability is zero. Instead, probability is smeared out as a density $f_{X} (x)$ : the height of a curve telling us how thickly probability piles up near each point.

WHY. With infinitely many outcomes we cannot sum blocks; a sum of infinitely many positive numbers would explode. The fix from calculus is to replace "add blocks" with "measure area under a curve." Area handles infinitely-thin slices gracefully. That is precisely why the tool we reach for here is the integral — it is the machine that adds up infinitely many infinitesimal slices.

PICTURE. The cyan curve is the density $f_{X}$ . We do not read probability off its height; we read it off the shaded area under it up to the marker $x$ .

$F_{X} (x) = \int_{- \infty}^{x} curve height at t f_{X} (t) width of a thin slice d t$

The $\int$ sign is a stretched "S" — the continuous cousin of $\sum$ ; it means "add the areas of all thin slices."
$t$ is a dummy sweeping variable running from far-left ( $- \infty$ ) up to the finish line $x$ . We use a different letter ( $t$ ) so we don't confuse the thing we sweep with the finish line $x$ that stays fixed.
$f_{X} (t) d t$ is the area of one skinny rectangle: height $f_{X} (t)$ times tiny width $d t$ .

Step 5 — Worked continuous example: Uniform(0,1) becomes a ramp

WHAT. Take $X \sim Uniform (0, 1)$ : equally likely anywhere in $[0, 1]$ , so its density is a flat rectangle of height $1$ over $[0, 1]$ and $0$ elsewhere. We slide the marker and read the accumulated area.

WHY. A flat density makes the area trivial to see: area under a rectangle of height $1$ from $0$ to $x$ is just base $\times$ height $= x \times 1 = x$ . This turns the abstract integral into a picture a child can measure with a ruler.

PICTURE. As the marker moves from $0$ to $1$ , the shaded rectangle's area grows linearly — that growing number is the CDF value, tracing a straight ramp from $0$ up to $1$ .

$F_{X} (x) = ⎩ ⎨ ⎧ 0 x 1 x < 0 (nothing to the left yet) 0 \leq x \leq 1 (area = base \times 1) x > 1 (all probability collected)$

Step 6 — The bridge: derivative undoes the integral

WHAT. Going up we accumulate: density → area → CDF via $\int$ . Going back down we ask "how fast is the accumulated area growing right now?" That rate of growth is the derivative $\frac{d}{d x} F_{X} (x)$ , and it returns the density.

WHY. This is the Fundamental Theorem of Calculus: integration and differentiation are inverse machines. We reach for the derivative specifically because it answers the question "at what rate is probability piling up at this exact spot?" — which is the very definition of density.

PICTURE. The steeper the ramp of $F_{X}$ , the taller the density $f_{X}$ underneath. A flat CDF ⇒ zero density; a fast-rising CDF ⇒ high density.

$f_{X} (x) = \frac{d}{d x} F_{X} (x) ⟺ F_{X} (x) = \int_{- \infty}^{x} f_{X} (t) d t$

The left arrow is "slope of the CDF"; the right is "area under the density." Same relationship, two directions.

Step 7 — Edge & degenerate cases (never leave the reader stranded)

WHAT. We check the extremes so no scenario surprises you.

WHY. A definition is only trustworthy if it behaves at the boundaries. Each edge below corresponds to a guaranteed property listed in the parent note.

PICTURE. The figure stacks four mini-panels: far-left tail, far-right tail, a point in the continuous case, and a subtraction on an interval.

$lim_{x \to - \infty} F_{X} (x) = 0 lim_{x \to + \infty} F_{X} (x) = 1$

Far left: the marker is before all outcomes, no probability collected ⇒ $0$ .
Far right: the marker is past every outcome, all probability collected ⇒ $1$ .

$P (X = x) = 0 (continuous X)$

A single point has zero width, so its slice has zero area — no probability at an exact point.

$P (a < X \leq b) = total up to b F_{X} (b) - total up to a F_{X} (a)$

Subtracting two running totals leaves exactly the strip between them. Always big minus small, or you get a meaningless negative.

Step 8 — Reading it backwards: the quantile (inverse CDF)

WHAT. So far we asked "given a value $x$ , how much probability is below it?" Now flip it: "given a probability $p$ , which value has that much below it?" That answer is the quantile function $F_{X}^{- 1} (p)$ , the subject of Quantile Functions.

WHY. This inverse is the engine behind Inverse Transform Sampling ( $X = F_{X}^{- 1} (U)$ turns uniform noise into any distribution), percentiles, and confidence thresholds. It is the same picture read along the other axis.

PICTURE. Enter on the vertical (probability) axis at height $p$ , walk right until you hit the curve, drop down — the landing spot on the horizontal axis is $F_{X}^{- 1} (p)$ .

$F_{X} (x) = p ⟺ x = F_{X}^{- 1} (p), median = F_{X}^{- 1} (0.5)$

The one-picture summary

Everything above compressed: density → (integrate/area) → CDF → (differentiate/slope) → density, with discrete sums as the block-stacking cousin and the quantile as the sideways read.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a number line and a marker you slide from left to right. As you slide, you sweep up probability like a snowplow. How much snow you have piled at any moment is the CDF. For separate outcomes (a die) the snow comes in equal blocks, so the pile grows in jumps — a staircase, and each riser's height is that outcome's chance. For a smeared-out quantity (a distance) the snow is measured as area under a density curve; a flat density gives a straight ramp. The steepness of your pile at any point tells you how thickly snow is falling there — that steepness is the density, which is why the derivative of the CDF gives back the density, and the integral of the density gives back the CDF. At the far left you have zero pile; at the far right you have all of it (one). To get the probability between two points, subtract the smaller pile from the bigger pile. And if instead you know how much snow you want and ask where the marker has to be, you read the same picture sideways — that is the quantile, the inverse CDF.

See also: Empirical Distribution · Kolmogorov-Smirnov Test · Expected Value and Variance · Copulas · Survival Analysis · back to Cumulative distribution functions.