Question

In the binomial replacement branching model with
P(S) = q + ps
, let  .

(a) Show that P[T=n] for n≥1 is  $p^{n-1} \cdot q$.

(b) Find P[T=n] for  .

Answer 1

In the binomial replacement branching model with
P(S) = q + ps
, let  
T = inf{n: Zn=0
.

(a) Show that P[T=n] for n≥1 is  .

b -ud

(b) Find P[T=n] for  

0 < ? = 7

In the binomial seplacement branching model with PO =+ps, let T = inf{n: 2:03. a). Grow that p{tin] for nslis phiq. Ans PS) = qtps. . T = inf{n: Zn zo} p=n) = P[inf {n: 2n 501 in), n 21, - P[Zn to happens first for the nith ly time] - Pfen =off P[Znzo, Em 1, Zn-2 = 1, .-- 2:=1]. = P[Zn =o] p[ 2n-=)... - P[2, +1] zis are independent. F. PCO).PO).. POD. = qtp.o) (q +P, 1) .... (@+P.1) (using PS) = 4+PCS) = .9 .{P+9) a... (P+9) (n-1) times

2. Find P[ian] assuming Z. =iso, Aly P[T =n91 20 =10) =P[inf Enizn =0} =n/zosiso] =P[2n 50;Zm1 =2, ..., Z=1, Z=1, Zori] = P(2no) p[287=1) .....P/22=1) P(2, -1) P(Zo-i) = 9(P+9)..... P+9) (P+9) * $ +P) = a (Pty) n/a +Pi).