Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 3 / b

Solution

Each completed visit to radius

r_{2}

begins a new radial excursion. By part (a), the conditional probability that the following excursion reaches radius

R

before radius

r_{1}

q_{R} = \frac{l o g ( r _{2} / r _{1} )}{l o g ( R / r _{1} )} .

τ_{2 k}

makes these trials independent with the same success probability. Consequently

N (R)

{1, 2, \dots}

q_{R}

P (N (R) > k) = (1 - q_{R})^{k} .

R \to \infty

q_{R} \to 0

and

q_{R} N (R) d Exp (1), q_{R} lo g R ⟶ lo g (r_{2} / r_{1}) .

\frac{N ( R )}{l o g R} d Exp (lo g (r_{2} / r_{1})) .

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