 Solution

ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2017/2/a/ii/solution

Differentiating the moment-generating function at zero gives

M_{X}^{'} (0) = E [X], M_{X}^{''} (0) = E [X^{2}] .

M_{X}^{''} (0) - (M_{X}^{'} (0))^{2}

. The assumed exponential integrability justifies these derivatives.

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