Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 3 / a

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- Solution a

Solution

The empty-bin indicators are not independent. For distinct

i, k

,

P (Z_{i} = Z_{k} = 1) = (1 - \frac{2}{n})^{m},

whereas

P (Z_{i} = 1) P (Z_{k} = 1) = (1 - 1/ n)^{2 m}

.

Each bin is empty precisely when all

m

balls avoid it, so

E Z_{i} = (1 - \frac{1}{n})^{m} .

The linearity of expectation does not require independence and gives

E Z = n (1 - \frac{1}{n})^{m} .

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