Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 14B / iii / Solution

Before the first product forms, the system is in either

E_{0}

or

C_{0}

. Product formation is possible only through

C_{0} \to E_{1}

, so the first-passage-time density is the outgoing probability flux

f (τ) = k_{2} C_{0} (τ) = \frac{k _{1} k _{2}}{λ _{+} - λ _{-}} (e^{- λ_{-} τ} - e^{- λ_{+} τ}), τ \geq 0.

Equivalently, the survival function is

E_{0} (τ) + C_{0} (τ)

and its negative derivative is this density.

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