 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ia/paper-1/12f/a/solution

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

ϕ (z) = \frac{1}{2 π} e^{- z^{2} /2}, z \in R .

It is nonnegative, and the Gaussian integral gives

\int_{- \infty}^{\infty} ϕ (z) d z = 1

M_{Z} (θ) = \int e^{θ z} ϕ (z) d z = e^{θ^{2} /2} .

Thus

E Z = M_{Z}^{'} (0) = 0

and

var (Z) = M_{Z}^{''} (0) - M_{Z}^{'} (0)^{2} = 1.

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