 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/m/m4/b/i/solution

∣ z ∣ < 14/5

(14 - 5 z)^{- 1/2} = \frac{1}{14} \sum_{n = 0}^{\infty} \frac{( n 2 n )}{4 ^{n}} (\frac{5}{14})^{n} z^{n} .

All coefficients are nonnegative, and

g (1) = 2/3 + 1/3 = 1

. Thus define

P (T = 0) = \frac{2}{3} + \frac{1}{14}, P (T = n) = \frac{1}{14} (n 2 n) (\frac{5}{56})^{n} (n \geq 1) .

These numbers sum to

1

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