 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-1/6h/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The joint density depends on the sample through

\sum_{i = 1}^{n} X_{i}^{2},

not merely through

\sum_{i} ∣ X_{i} ∣

. For example, after padding with zeros when

n > 2

, take

x = (1, 1, 0, \dots, 0), y = (2, 0, 0, \dots, 0) .

Both samples have sum of absolute values

2

, but

\frac{p _{σ} ( x )}{p _{σ} ( y )} = exp (\frac{1}{σ ^{2}}),

which depends on

σ

. The likelihood-ratio necessary condition for sufficiency fails, so

i = 1 \sum n ∣ X_{i} ∣ is not sufficient .

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