 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/m/m4/a/i/solution

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

for real

t

where this expectation is finite. Usually one requires finiteness on a neighbourhood of zero when using it to characterize a distribution. The characteristic function is

φ_{X} (t) = E [e^{i tX}], t \in R .

It always exists because the integrand has absolute value one. In both definitions the expectation is taken with respect to the distribution of

X

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