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November 13th, 2016, 05:34 AM   #1
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Linear birth and death process

X(t) - Is the number of people at time t >= 0, which is modelled by a linear birth and death process

y - birth rate
u - death rate

X(0) = a - Initial population size

$\displaystyle q(t) = P(X(t) = 0 | X(0) = 1) $
$\displaystyle q_j(t) = P(X(t) = 0|X(0) = j)$

How can I prove that
$\displaystyle
q_j(t) = (q(t))^j $, j>=2
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November 13th, 2016, 01:26 PM   #2
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Look at the problem as j independent events, so you can use product rule for probabilities of independent random variables.
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