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Extinction probability of a linear birth-death process

For individual birth and death rates

λ, μ

and one initial individual, the finite-time extinction probability satisfies

h^{'} = (h - 1) (λh - μ), h (0) = 0.

When

λ \neq = μ

,

h (t) = \frac{μ ( 1 - e ^{- (λ - μ) t} )}{λ - μ e ^{- (λ - μ) t}} .

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