 Solution

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

X

M_{X} (t) = E [e^{tX}]

for real

t

where the expectation is finite. One may regard it as an extended nonnegative function elsewhere. In particular

M_{X} (0) = 1

; finiteness on a neighborhood of zero is the usual hypothesis for recovering moments and using its local expansion.

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