In the spreading of a rumour, each person who is told the rumour has probability p of believing it andonly those who believe the rumour will pass it on. Assume that all individuals behave independently ofeach other and the rumour is never told to someone who has already heard it. Assume also that eachperson who is told the rumour and believes it on the n-th day will pass the rumour on to two other peopleon the (n + 1)-st day.Suppose that the rumour originates on day 1 from a single individual telling the rumour to two otherpeople. Let Zo = 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day nand believe it.(i) Explain why (Zn)nzo is a branching process.(ii) Find the probability mass function and the probability generating function of Z₁.(iii) What is the probability that the spread of the rumour ultimately ceases?(iv) For what values of p is the spread of the rumour certain to cease eventually?(v) For n ≥ 1 show thatP(Zn = 2") = p2¹ +22+ +2″

Given, and for denote by the number of people who are told the rumour on day n and believe it.

Answered: In the spreading of a rumour, each…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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