4. Let Zn be the size of the nth generation of a branching process, and assume Zo = 1. Find an expression for the generating function Gn of Zn, in the cases when Z₁ has generating function given by: (a) G(s) = 1-a(1-s), 0 1. Calculate the answer explicitly.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 43E
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4. Let Zn be the size of the nth generation of a branching process, and assume Zo = 1. Find an
expression for the generating function Gn of Zn, in the cases when Z₁ has generating function given
by:
(a) G(s) = 1-a(1-s), 0 <a, ß < 1.
(b) G(s) = f-¹{P(f(s))}, where P is a probability generating function, and f is a suitable function
satisfying f(1) = 1.
(c) Suppose in the latter case that f(x) = xm and P(s) = s{y-(y-1)s)-¹ where y > 1. Calculate
the answer explicitly.
Transcribed Image Text:4. Let Zn be the size of the nth generation of a branching process, and assume Zo = 1. Find an expression for the generating function Gn of Zn, in the cases when Z₁ has generating function given by: (a) G(s) = 1-a(1-s), 0 <a, ß < 1. (b) G(s) = f-¹{P(f(s))}, where P is a probability generating function, and f is a suitable function satisfying f(1) = 1. (c) Suppose in the latter case that f(x) = xm and P(s) = s{y-(y-1)s)-¹ where y > 1. Calculate the answer explicitly.
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