 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-41/1/d/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Four unsuccessful attempts have conditional probability

\prod_{j = 1}^{4} q_{j} (y)

. The outcome is then unobserved, so the observed-data likelihood integrates over it:

L_{i} = \int_{- \infty}^{\infty} \frac{1}{σ} ϕ (\frac{y - μ _{i}}{σ}) j = 1 \prod 4 \frac{e ^{η_{j} + θ y}}{1 + e ^{η_{j} + θ y}} d y .

This is a probability, whereas a responder contributes a joint density in the observed outcome. The sequential response selection model keeps outcome-dependent nonresponse in the likelihood; evaluating the product at an imputed mean would not be this likelihood.

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