 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/27j/d/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Z = e^{i u X}

. Then

∣ Z ∣ = 1

E [∣ Z - E Z ∣^{2}] = E ∣ Z ∣^{2} - ∣ E Z ∣^{2} = 1 - ∣ μ_{X} (u) ∣^{2} = 0.

Thus

Z = E Z

almost surely. Its constant value has modulus one, so it equals

e^{i θ}

for some

θ \in R

u X \in θ + 2 π Z almost surely .

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