Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 7 / c / i

Table of contents
- Solution i

Solution

Summing the probability mass function of the binomial distribution up to the largest permitted integer gives

0 \leq k \leq α n \sum (k n) p^{k} (1 - p)^{n - k} = P (S_{n} \leq ⌊ α n ⌋) = P (S_{n} / n \leq α) .

The floor is implicit in the original integer-indexed sum.

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