Foundation lesson
Density is not probability
A density is probability per unit length. A probability belongs to an interval, and it is the area under the curve over that interval.
Written for this edition, not translated from Einstein.
What does the height of a probability curve mean?
Mark where 100 particles end up on a ruler divided into 2 μm bins. If 30 land in one bin, that bin holds a probability of about 0.3, and the density there is 0.3 divided by 2 μm, or 0.15 per micrometre. Measured in metres the same density is 150,000 per metre, while the bin still holds 0.3: the height depends on the unit of length, and the probability does not.
The probability of the interval from a to b is the integral of the density over that interval.
Under a smooth curve, any single exact position has probability zero, since an interval of zero width has no area. At the starting instant of the Brownian problem every particle sits at one point. No curve can draw that: the whole probability, 1, sits on the single point, and a curve appears only once the particles have begun to spread.
Worked example: A flat density across four micrometres
- Spread the probability evenly over 4 μm. The density is 1 divided by 4 μm, or 0.25 per micrometre.
- The area over the whole 4 μm is 0.25 per μm × 4 μm = 1.
- A 1 μm piece has probability 0.25 × 1 = 0.25, and a 2 μm piece has 0.5.
- Squeeze the same probability into 2 μm and the height doubles to 0.5 per micrometre, while the total stays 1. The height was never a probability.
The height moves, the area does not
One bin, holding 30 of the 100 particles. Its height is written twice, in two units of length; what it encloses is written once.
Change the ruler from micrometres to metres and the height is multiplied by a million. The bin still holds 0.3 of the particles. A number that moves when you change the ruler was never the probability.
The same the other way round: one whole unit of probability, spread over 4 μm and then squeezed into 2.
What it shows, in words
The first drawing is one 2 micrometre bin holding 30 of 100 particles. Measured per micrometre its height is 0.15; measured per metre the same height is 150,000. Nothing about the particles changed between those two numbers, only the ruler. The area the bar encloses is 0.3 under either, and that is the bin's probability.
The second drawing is one whole unit of probability twice over. Spread across 4 micrometres it stands 0.25 per micrometre tall; squeezed into 2 it stands 0.5, exactly twice as tall over half the width. The two rectangles enclose the same area, 1, because it is the same probability. The height was never the probability; the area always was.
Where this lesson stops
The height of the curve is a density, a probability per unit length. Only the area over an interval is a probability.
Going further: One curve from different causes, and a curve turned around
A distribution can be fixed without its causes being fixed. The spreading of Brownian particles has one curve for each value of D, and in the Brownian paper D = RT/(N · 6πkP), with k the liquid's viscosity and P the particle's radius. At a fixed temperature and viscosity, a radius twice as large with an N half as large gives the same D, so it gives the same curve. No number of observed displacements can tell the two apart: the samples depend only on the product N·P, so only that product can be read from them.
Open the foundation: Two measurements, two unknowns
A change of variable reshapes a distribution. If an estimate of D is turned into an estimate of N by dividing a constant by it, the distribution of N is not the distribution of D relabelled. Equal steps in D become unequal steps in N, so the density is stretched in some places and squeezed in others, while each pair of matching intervals keeps the same probability.
- Suppose an estimate of D comes out as 4 or 6 with equal chance. Its average is 5, the true value, so it has no bias.
- Estimate N as 60 ÷ D. The two outcomes give 15 and 10, whose average is 12.5, but 60 ÷ 5 is 12. The turned estimate is too large by about 4 per cent on average, though nothing about the measurement changed.
- An interval survives the turn with its ends swapped: D between 4 and 6 is the same event as N between 10 and 15, so the two have the same probability.
- For an estimate built from q = 100 squared Gaussian displacements the same effect is smaller: the turned estimate is too large on average by the factor q/(q − 2) = 1.0204, about 2 per cent.
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