 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability-resit/8/b/ii/solution

Let

I_{i}

indicate that student

i

stays awake. These indicators have independent Bernoulli distributions with parameter

p

, and

W = \sum_{i = 1}^{M} I_{i}

G_{M} (u) = \sum_{m = 0}^{\infty} e^{- λ} \frac{λ ^{m}}{m !} u^{m} = e^{λ (u - 1)} .

G_{W} (s) = G_{M} (1 - p + p s) = e^{λ p (s - 1)} .

W \sim Poisson (λ p), P (W = w) = e^{- λ p} \frac{( λ p ) ^{w}}{w !}, w \geq 0.

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