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Math Help - Age dependent branching process

  1. #1
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    Age dependent branching process

    Let Z_n be the size of the nth generation in an age-dependent branching process Z(t), the lifetime distribution of which is exponential with parameter \lambda. If Z(0) = 1, show that the probability generating function G_t (s) of Z(t) satisfies

     \frac{\partial}{\partial t} G_t(s) =  \lambda {G(G_t(s)) - G_t(s)}

    i get:

    from a theorem:

    G_t(s) = \int^t_0 G(G_{t-u})f_T(u)\ du + \int^\infty_t sf_T(u) \ du

    differentiating with respect to t

    \frac{\partial}{\partial t} G_t(s) = G(G_0(s))f_T(t) - sf_T(t)

    however i am not sure this right as the Generating functions are confusing and i am uncertain as where to go next.
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    Re: Age dependent branching process

    Hello,

    When you're differentiating the first integral, the constant is when u=t, and the part to be differentiated is when u=0.

    This would rather give \frac{\partial}{\partial t} G_t(s)=G(G_t(s))f_T(0)+sf_T(t)

    and since f_T is the pdf of an exponential, f_T(0)=\lambda. But as to get sf_T(t)=G_t(s), I don't know, sorry...
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