Correlation between two bin counts when items are sorted at random

Question

Each of 100 items is placed independently into one of 5 bins, all bins equally likely. Let N1 and N2 be the counts in bins 1 and 2. What is the correlation between N1 and N2?

Accepted Answer

Every bin count competes for the same 100 items, so any two are negatively correlated. Build both from per-item counters to get the pieces. Let p = 	frac15 be each bin's chance. For counts over n = 100 items: - Covariance of two different bins: \mathrm{Cov}(N1, N2) = -n\,p\,p = -100\cdot	frac15\cdot	frac15 = -4. - Variance of one bin count: \mathrm{Var}(N1) = n\,p(1-p) = 100\cdot	frac15\cdot	frac45 = 16, so each standard deviation is 4. Correlation. ho = \frac{\mathrm{Cov}(N1, N2)}{\mathrm{SD}(N1)\,\mathrm{SD}(N2)} = \frac{-4}{4 	imes 4} = -\frac14 = -0.25. There is a clean shortcut w…

Correlation between two bin counts when items are sorted at random

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