 Solution

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

E T = g^{'} (1)

and

E [T (T - 1)] = g^{''} (1)

. Here

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

Thus

E T = 5/54

and

E [T (T - 1)] = 25/324

. Adding the first moment gives

E T^{2} = 55/324

, hence

E T = \frac{5}{54}, Var (T) = \frac{55}{324} - \frac{25}{2916} = \frac{235}{1458} .

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