Accumulation, not decoration
Every sliver of area under the density gets added to a running total. That running total is the cumulative curve — it only ever climbs, and it starts at 0 and ends at 1.
Try: drag all the way right → F(x) hits 1.000
The density curve and the cumulative curve are the same information twice. Drag the line: the area you sweep out under the density above is exactly the height the cumulative curve reaches below.
Every sliver of area under the density gets added to a running total. That running total is the cumulative curve — it only ever climbs, and it starts at 0 and ends at 1.
The dashed tangent on the lower panel has slope f(x). Tall density → steep climb. Density near zero → the cumulative curve goes flat, even though probability is never lost.
A count has no area to sweep — it has bars to add. So the cumulative curve is a staircase, and P(X = k) shows up as the height of one step rather than a slope.
| Quantile p | x with F(x) = p | density f(x) |
|---|