 Solution

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

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

1

and

2

. Moment-generating functions finite near zero determine the distribution, so

Y

has that mixture law. Consequently

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

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