 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/ia/paper-2/9f/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-07

Finiteness of the moment-generating function on an open neighborhood of zero gives its differentiability there. Using the permitted interchange of differentiation and expectation,

M_{X}^{'} (t) = E [X e^{tX}], M_{X}^{''} (t) = E [X^{2} e^{tX}] .

Hence

M_{X} (0) = 1

M_{X}^{'} (0) = μ

and

M_{X}^{''} (0) = s^{2}

. The second-order Taylor expansion with its Peano remainder gives

M_{X} (t) = 1 + μ t + \frac{1}{2} s^{2} t^{2} + o (t^{2}) .

Here

s^{2}

is the raw second moment, not the variance:

Var (X) = s^{2} - μ^{2}

. This distinction matters before specializing to a centered variable.

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