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In the spreading of a rumour, each person who is told the rumour has probability p of believing it and only those who believe the rumour will pass it on. Assume that all individuals behave independently of each other and the rumour is never told to someone who has already heard it. Assume also that each person who is told the rumour and believes it on the n-th day will pass the rumour on to two other people on the (n + 1)-st day. Suppose that the rumour originates on day 1 from a single individual telling the rumour to two other people. Let Zo= 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day n and believe it. (i) Explain why (Zn) n20 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 that P(Zn = 2) = p²¹ +2²++2"

In the spreading of a rumour, each person who is told the rumour has probability p of believing it and only those who believe the rumour will pass it on. Assume that all individuals behave independently of each other and the rumour is never told to someone who has already heard it. Assume also that each person who is told the rumour and believes it on the n-th day will pass the rumour on to two other people on the (n + 1)-st day. Suppose that the rumour originates on day 1 from a single individual telling the rumour to two other people. Let Zo= 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day n and believe it. (i) Explain why (Zn) n20 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 that P(Zn = 2) = p²¹ +2²++2"

MATLAB: An Introduction with Applications
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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In the spreading of a rumour, each person who is told the rumour has probability p of believing it and only those who believe the rumour will pass it on. Assume that all individuals behave independently of each other and the rumour is never told to someone who has already heard it. Assume also that each person who is told the rumour and believes it on the n-th day will pass the rumour on to two other people on the (n + 1)-st day. Suppose that the rumour originates on day 1 from a single individual telling the rumour to two other people. Let Zo = 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day n and 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 that P(Zn = 2") = p2¹ +22+ +2″
Transcribed Image Text:In the spreading of a rumour, each person who is told the rumour has probability p of believing it and only those who believe the rumour will pass it on. Assume that all individuals behave independently of each other and the rumour is never told to someone who has already heard it. Assume also that each person who is told the rumour and believes it on the n-th day will pass the rumour on to two other people on the (n + 1)-st day. Suppose that the rumour originates on day 1 from a single individual telling the rumour to two other people. Let Zo = 1 and for n ≥ 1 denote by Z₁, the number of people who are told the rumour on day n and 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 that P(Zn = 2") = p2¹ +22+ +2″
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