Statistics
Probability Density
For a continuous quantity no single value has any probability at all: only stretches do. A density is the height whose area is that probability, so every question becomes an integral. Drag the interval on the curve, or drag a region across a two-dimensional density, and read the area you enclosed.
P(a ≤ X ≤ b)—
Total area—
Tallest the density gets—
Width b − a—
In one dimension, drag either end of the shaded stretch. In two, drag the circle around. The number is always the same thing: the amount of density you have enclosed.
What to observe
- Press Shrink it to a point. The shaded area collapses and the probability goes to zero, even though the density there is perfectly healthy. For a continuous quantity, P(X = c) = 0 always, and that is not a paradox: it takes a stretch to hold any probability.
- Pick narrow. The curve shoots past height 4, which no probability may do. A density is not a probability; it is probabilityper unit of x, and only its area has to behave.
- The curve underneath is the running total, the CDF. The shaded area upstairs is exactly the vertical step F(b) − F(a) downstairs. Two pictures of one number.
- Try decay and slide the stretch to the right. Equal widths keep giving smaller probabilities, because the density is what changes, never the width.
- Switch to two dimensions. Now the density is a height over the plane, the region is a patch of ground, and the probability is the volume standing on it. Every one-dimensional idea survives with an extra integral.
- In two dimensions, watch the dots as you move the circle. The share of dots caught inside tracks the integral, which is exactly what makes Monte Carlo work.
Shortcuts: space run/pause · s step · r reset · f fullscreen