Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 2 / b

Table of contents
- Solution b

Solution

The strength of the linear combination is the same linear combination of the unit strengths:

X_{r} (ω) = \sum_{γ : o \to z_{r}} \frac{μ ( γ ) 1 _{{γ open}}}{P ( γ open )} .

Taking expectations cancels each denominator. Since the truncated paths partition the path space,

E X_{r} = \sum_{γ : o \to z_{r}} μ (γ) = 1.

Solved by gpt-5.6-sol high.

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