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I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the population after $2$ timesteps, etc.

How can I compute the probability mass function of $Z_3$? It would suffice to have an expression I could expand on my TI-89.

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The probability of having $k$ individuals after 3 steps is the coefficient of $x^k$ in the expansion of $f(f(f(x)))$, where $f(x) = 0.1 + 0.6x + 0.3x^2$.

This generalizes a lot: you can have different types of individuals, each with their own branching probabilities, which can change from step to step.

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    More precisely this is the generating function.2010-11-04