Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 6C / b

Table of contents
- Solution b

Solution

Multiply the birth-death master equation by

n

and sum. Index shifts give

\overset{μ}{˙} = (B - D) μ .

For the second moment, a birth changes

n^{2}

by

2 n + 1

and a death changes it by

- 2 n + 1

, so

\frac{d}{d t} E [N^{2}] = 2 (B - D) E [N^{2}] + (B + D) μ .

Since

σ^{2} = E [N^{2}] - μ^{2}

,

(σ^{2})^{'} = 2 (B - D) σ^{2} + (B + D) μ .

Together these are the moment equations of a linear birth-death process.

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