Results 1 to 2 of 2

Thread: Age dependent branching process

  1. #1
    Member
    Joined
    Aug 2010
    Posts
    77

    Age dependent branching process

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

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

    i get:

    from a theorem:

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

    differentiating with respect to t

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

    however i am not sure this right as the Generating functions are confusing and i am uncertain as where to go next.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6

    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 $\displaystyle \frac{\partial}{\partial t} G_t(s)=G(G_t(s))f_T(0)+sf_T(t)$

    and since $\displaystyle f_T$ is the pdf of an exponential, $\displaystyle f_T(0)=\lambda$. But as to get $\displaystyle sf_T(t)=G_t(s)$, I don't know, sorry...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Branching Process
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Jul 21st 2010, 03:37 PM
  2. Branching Process
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Apr 20th 2010, 05:47 AM
  3. Age-dependent branching process
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Apr 12th 2010, 07:49 PM
  4. Branching Process
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Mar 27th 2010, 09:09 PM
  5. Branching Process.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 26th 2009, 07:09 AM

Search Tags


/mathhelpforum @mathhelpforum