 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/m/m1/a/solution

For rate

λ > 0

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

This function is finite near zero, so its third derivative there equals the third moment. Therefore

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

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