 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/8/b/iii/solution

G_{N}^{'} (s) = p G_{N} (s)^{2} + 2 p s G_{N} (s) G_{N}^{'} (s) .

s = 1

G_{N} (1) = 1

E [N] = G_{N}^{'} (1) = \frac{p}{1 - 2 p} .

The derivative is finite because

p < 1/2

and the explicit square root has no singularity at

s = 1

. Alternatively, the expected population in generation

t

(2 p)^{t}

, so the expected number of splits in that generation is

p (2 p)^{t}

. Summing this nonnegative geometric series gives the same answer without initially assuming a finite expected value.

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