 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/probability/7/b/v/solution

E N_{n} = n p_{n}

. The variance of a sum and the negative covariance just proved give

var (N_{n}) = n p_{n} (1 - p_{n}) + n (n - 1) (q_{n} - p_{n}^{2}) \leq n p_{n} (1 - p_{n}) \leq \frac{n}{4} .

P (\frac{N _{n}}{n} - p_{n} > ϵ) \leq \frac{var ( N _{n} )}{n ^{2} ϵ ^{2}} \leq \frac{1}{4 n ϵ ^{2}} ⟶ 0.

p_{n} \to e^{- 1}

N_{n} / n

e^{- 1}

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