Statistics
Probability Density
Drop grains of sand where a random quantity lands. They pile up into a shape; divide each pile by its width and the shape becomes the density curve. Then the curve is a sandpile: the share of grains over a stretch is the area there, the total is always one, and a single exact point holds none.
grains: one outcome eachdensity curve f(x)the stretch from a to b
In chapter 3, drag the two edges of the stretch
Now—
—
Chapter
Where the grains fall
squeezedspread
Now
grains—
grains in the stretch—
area over the stretch—
highest point of f—
What to observe
- The first grains land almost anywhere. Only after hundreds does the pile take a shape, and that shape is the same every time.
- Halving the bins halves how many grains each catches, so raw counts would shrink to nothing. Dividing by the width keeps the heights steady, and they settle onto one curve.
- Drag the stretch: the share of grains inside it and the area under the curve there are the same number. That is all a density is for.
- Squeeze the spread: the curve climbs past 1, yet no grain is added, so the area stays exactly 1. A density is not a probability and can be any height.
- Shrink the stretch around one point: the grains inside run out and the probability goes to 0, while probability ÷ width settles on the height f(x).
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