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Moments of a batch-birth linear-death process

For

m = E N

and

v = var N

, the Markov jump-process generator applied to

n

and

n^{2}

gives

m^{'} = kλ - β m, v^{'} = k^{2} λ + β m - 2 β v .

Consequently

m ⟶ \frac{kλ}{β}, v ⟶ \frac{k ( k + 1 ) λ}{2 β} .

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