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

Table of contents
- Solution d

Solution

Exactly one lowest-numbered ball is selected in each occupied bin, so

\sum_{j = 1}^{m} α_{j} (x) = n - f (x) \leq n, \sum_{j = 1}^{m} α_{j} (x)^{2} \leq n .

Part c supplies the corresponding one-sided coordinate certificate. The product-space entropy method for certifiable functions states that a function with such a certificate of squared size at most

v

has both centered tails bounded by

e^{- t^{2} / (2 v)}

. Taking

v = n

gives

P (Z - E Z \geq t) \leq e^{- t^{2} / (2 n)}, P (Z - E Z \leq - t) \leq e^{- t^{2} / (2 n)} .

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