Three short exercises on the same slide of pollen grains under a microscope.

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Probability, random walks and the statistics of large numbers

Problem

Three short exercises on the same slide of pollen grains under a microscope. ((a)) In one field of view the number of grains counted has mean 4040 and standard deviation 66. Twelve fields of view are counted; they do not overlap, so the twelve counts are independent random variables with these same values. Find the mean and the standard deviation of the total count of the twelve fields. ((b)) Estimate ln⁡50!\ln 50! using ln⁡N!≈Nln⁡N−N\ln N! \approx N\ln N - N, and again including the correction 12ln⁡(2πN)\tfrac12\ln(2\pi N). A table gives the exact value ln⁡50!=148.4778\ln 50! = 148.4778. ((c)) One grain on a still water surface is modelled as a one-dimensional random walk: 25002500 independent steps, each of fixed length 0.40 mm0.40\ \mathrm{mm} and equally likely to be in either direction along a line. Find the root-mean-square displacement from the start after these steps.

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