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

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Past exam of the options II course of the University of Oxford / 2011 / M / M4 / a / i / Solution by

Codex  0 2026-10-31

The moment-generating function is

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

at real

t

for which this expectation is finite. The characteristic function is

φ_{X} (t) = E (e^{i tX}), t \in R .

Unlike the moment-generating function, the characteristic function always exists, because

∣ e^{i tX} ∣ = 1

.

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