Foundation · Explanatory preview
Density is not probability
A density measures probability per unit of position. Probability belongs to an interval and is its area under the curve.
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What does the height of a probability curve mean?
Divide a ruler into bins. The fraction of observations in a bin estimates its probability. Dividing that fraction by the bin width estimates a density. Changing the ruler from metres to micrometres changes the numerical density but not the probability of the same physical interval.
The probability of the interval from a to b is the integral of the density over that interval.
A continuous distribution gives probability zero to any single exact position. At the starting time the ideal point source is instead an atom of probability one, not a finite curve with infinite height.
One worked example
A uniform density of 0.25 per micrometre on a four-micrometre interval has total probability one. A one-micrometre subinterval has probability 0.25; a two-micrometre subinterval has probability 0.5.
A stopping point: Always distinguish curve height, interval area and an individual observation.
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