Past exam of the stochastic modelling of biological processes course of the University of Oxford / 2023 / 1 / b

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- Solution b

Solution

At the empty Markov chain state, only the constant-rate production chemical reaction is possible. Thus the first-passage time

τ_{1}

has an exponential distribution of rate

2

:

P (τ_{1} > t) = e^{- 2 t}, t \geq 0.

Its expected value can be obtained by integrating this survival function, and its second moment by integrating

2 t

times the survival function:

E τ_{1} = \int_{0}^{\infty} e^{- 2 t} d t = \frac{1}{2}, E τ_{1}^{2} = 2 \int_{0}^{\infty} t e^{- 2 t} d t = \frac{1}{2} .

Consequently the requested expected value and variance are

E τ_{1} = \frac{1}{2}, Var (τ_{1}) = \frac{1}{4} .

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