page392 Q2 Branching processe

Let Z1, Z2, . . . , ZN describe a branching process in which each parent has j offspring with probability pj. Find the probability d that the process dies out if

Each indivial 1,2,…..\({ Z }_{ n-1 }\) has same family distribution with ramdom size Z0,Z1,Z2,… Zi, each has constant probability pi, \(i\quad \in \quad [0,\quad j]\). Zi represnt i child with probability p(Zi). Assume t goes to infinity for process dies out, each distribution of Zn satisfys ordinary generation function ${ h }(z)={ 0 }+{ p }{ 1 }z+_{ 2 }{ z }^{ 2 }+…$ for each branch process Zi.

For the probability generation function d, ${ d }={ 0 }+{ 1 }d+_{ 2 }{ d}^{ 2 }+…$

  1. p0 = 1/2, p1 = p2 = 0, and p3 = 1/2.

\({ d }=\quad \frac { 1 }{ 2 } \quad +\quad 0*d\quad +\quad 0*d^{ 2\quad }+\quad \frac { 1 }{ 2 } *d^{ 3\quad }\)

  1. p0 = p1 = p2 = p3 = 1/4.

\({ d }=\quad \frac { 1 }{ 4 } \quad +\quad \frac { 1 }{ 4 } *d\quad +\quad \frac { 1 }{ 4 } *d^{ 2\quad }+\quad \frac { 1 }{ 4 } *d^{ 3}\)

  1. p0 = t, \(p1=1-2t\), p2 = 0, and p3 = t, where \(t\quad\le\quad\frac{ 1 }{ 2 }\).

\({ d }=\quad t\quad +\quad (1-2t)*d\quad +\quad 0*d^{ 2\quad }+\quad t*d^{ 3\quad }\quad (t\le 0.5)\)