From PDF to CDF
The continuous version of the previous demo. Now $X$ takes values on a continuum, not a discrete set. Probability is described by a density $f(x)$ — itself not a probability. A single point has probability zero; only intervals carry probability, given by area:
\[P(a < X \leq b) = \int_a^b f(t)\,dt.\]The CDF is built the same way as before, with the running sum of point masses replaced by a running integral of density:
\[F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t)\,dt.\]Pick a distribution, tune its parameters, then sweep the current $x$. The blue area under the density to the left of $x$ equals the height of the CDF at $x$ — the same probability, viewed two ways.
Interactive
What it shows
Density vs. probability. $f(x)$ measures how concentrated probability is around $x$, in units of probability per unit of $x$. It can exceed 1 — try a Normal with $\sigma = 0.2$. Probability itself only emerges by integrating over a region of positive width.
The CDF stays bounded. $F(x)$ runs from 0 to 1, monotone non-decreasing, smooth wherever $f$ is finite. No jumps in the continuous case — those required point masses.
Discrete to continuous. Compared to the PMF demo, the lollipop spikes have melted into a smooth density and the CDF staircase has straightened into a continuous curve. The accounting principle is unchanged: cumulative probability at $x$ is whatever sits “to the left” of $x$, just measured by area instead of by counting masses.