 Solution

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

M_{Y} (t) = \frac{1}{2} \frac{1}{1 - t} + \frac{1}{2} \frac{2}{2 - t} .

It is a half-and-half mixture of exponential distributions with rates one and two. Uniqueness of a moment-generating function finite near zero identifies the law. Therefore

P (Y \leq 1) = \frac{1}{2} (1 - e^{- 1}) + \frac{1}{2} (1 - e^{- 2}) = 1 - \frac{1}{2} (e^{- 1} + e^{- 2}) .

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