 Solution

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

g^{'} (z) = \frac{5}{2} (14 - 5 z)^{- 3/2}, g^{''} (z) = \frac{75}{4} (14 - 5 z)^{- 5/2} .

Therefore

E T = g^{'} (1) = \frac{5}{54}, E [T (T - 1)] = g^{''} (1) = \frac{25}{324} .

Using the relation between the second factorial moment and variance,

Var (T) = g^{''} (1) + g^{'} (1) - g^{'} (1)^{2} = \frac{235}{1458} .

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