 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-3/18h/c/solution

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

Bayes' formula divides the point-null contribution to the mixture density by the full marginal density:

q (\overline{x}) = \frac{ϕ _{1/ n} ( x )}{ϕ _{1/ n} ( x ) + ϕ _{τ^{2} + 1/ n} ( x )} .

Since

\frac{ϕ _{τ^{2} + 1/ n} ( x )}{ϕ _{1/ n} ( x )} = \frac{1}{1 + n τ ^{2}} exp [\frac{n ^{2} τ ^{2} x ^{2}}{2 ( 1 + n τ ^{2} )}],

q (\overline{x}) = {1 + \frac{1}{1 + n τ ^{2}} exp [\frac{n ^{2} τ ^{2} x ^{2}}{2 ( 1 + n τ ^{2} )}]}^{- 1} .

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