 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/m/1/a/solution

λ > 0

, direct integration gives the moment-generating function

M_{X} (t) = \int_{0}^{\infty} e^{t x} λ e^{- λ x} d x = \frac{λ}{λ - t}, t < λ .

It is finite in a neighborhood of zero, so its derivatives there give moments. Differentiating three times gives

M_{X}^{(3)} (t) = 6 λ / (λ - t)^{4}

, hence

E X^{3} = M_{X}^{(3)} (0) = 6/ λ^{3} .

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