Question: Branching process to model rumor spread We consider a branching process (Zn)nEN given by the following dynamics: 0 With probability 1 p, an individual has

Branching process to model rumor spread

We consider a branching process (Zn)nEN given by the following dynamics: 0 With probability 1 p, an individual has 0 offspring. 0 With probability p, the individual has a positive number of offspring decided as described below. 0 The individual performs an experiment with failure probability q and success probability 1 q, repeatedly until a success occurs. The number of offspring is given by the number of trials required. (a) Determine the osp1'ing distribution (1;, = P(X = k). (1)) Provide a condition (on p, q) in order for the process to be supercritical. (c) In the supercritical case, determine the extinction probability as a function of p and q. (d) A model for spreading a rumor is designed as follows: any individual who hears the rumor will decide to spread it or not with probability 1/2. If they decide to spread it1 every time they tell the rumor to someone there is a probability 1/3 that they'll decide not to spread it further. One person starts a rumor. What is the probability that the rumor will eventually die out ? (e) What is the expected number of people who have heard the rumor after 3 steps of spread

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