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 Incoming links: Squared discrepancy of a digit frequency

Past exam of the physics A course of the University of Cambridge / 2013 / Paper 1 / Section A / 4 / Solution 2026-10-20

For the squared discrepancy of a digit frequency, write

d_{N}^{2} = (f_{0} - 0.1)^{2}

. The plotted ordinates are

1 0^{- 2}

4.00 \times 1 0^{- 4}

1.69 \times 1 0^{- 4}

4.90 \times 1 0^{- 5}

3.60 \times 1 0^{- 5}

and

9.00 \times 1 0^{- 6}

in increasing

N

. Under an independent, equiprobable-digit model, the zero count is binomial and

E [d_{N}^{2}] = Var (f_{0}) = \frac{0.1 ( 0.9 )}{N} = \frac{0.09}{N} .

(1)

Thus the expected log-log slope is $- 1$ . If “relative” means division by the expected fraction before squaring, every ordinate is multiplied by 100 and the prediction becomes

9/ N

; the slope is unchanged. These are nested prefixes of one digit sequence, so the points are correlated, rounded observations, not independent variance estimates. Their rough trend supports a reciprocal-power scale but does not prove independence or that

π

is a normal number.

Figure 1.
Original log-log plot of squared digit-frequency discrepancies with the independent-digit expectation
. **Details**.

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