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For Questions 1-4, consider the following scenario. Suppose thatthere are N individuals in a population, labeled 1, 2, . . . , N.Suppose that individuals 1, 2, . . ...

Question

For Questions 1-4, consider the following scenario. Suppose thatthere are N individuals in a population, labeled 1, 2, . . . , N.Suppose that individuals 1, 2, . . . , A are infectedwith an infectious disease at the end of day 0. On day 1 each ofthem will choose, independently and uniformly at random, anindividual in the population (possibly themselves)and infect them if they are not already infected. For 1 â‰¤ i â‰¤ A,let Xi be the individualthat person i tries to infect on day 1, so that X1, X

For Questions 1-4, consider the following scenario. Suppose that there are N individuals in a population, labeled 1, 2, . . . , N. Suppose that individuals 1, 2, . . . , A are infected with an infectious disease at the end of day 0. On day 1 each of them will choose, independently and uniformly at random, an individual in the population (possibly themselves) and infect them if they are not already infected. For 1 â‰¤ i â‰¤ A, let Xi be the individual that person i tries to infect on day 1, so that X1, X2, . . . , XA are i.i.d, uniformly distributed on 1, 2, . . . , N. 1. (10pts) For A + 1 â‰¤ j â‰¤ N, let Zj be the indicator that person j gets infected on day 1, that is Z j = (1 if person 0 otherwise. j gets infected on day 1 . Explain why Z j = 1 - Aâˆi=1 1(Xi 6= j). 2. For Zj defined in Problem 1 1. (4pts) Compute E(Zj). 2. (6pts) For j 6= k, compute E(ZjZk) 3. (10pts) Let Y be the number of individuals infected by the end of day 1, so that Y = A + Nâˆ‘ j=A+1 Z j. Suppose e > 0 and use Chebyshevâ€™s inequality to find an upper bound for P(jY - E(Y)j > e). Your answer should be in terms of the values you computed in Problem 2. . 4. (10pts) Suppose that N and A tend to infinity in such a way that A/N ! a 2 (0, 1). Show that, for e > 0, P(jY - E(Y)j > eN) ! 0.

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