Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, acoin having probability p, 0 < p < 1, of landing heads, is flipped. If heads appear, then a ball ischosen at random from the urn and is replaced by a white ball; if tails appear, then a ball is chosenfrom the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urnafter the n stages.Q6(i.) Is {Xn, n 2 0} a Markov Chain? If so, explain why.Q6(ii.) Compute the transition probabilities Pij: (Hint: At some stage, suppose there are i whiteballs (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the nextstage.)Q6(iii.) Let N2. Then the states are 0, 1, 2 and the transition probability matrix is given by를 + 올 끌Find the proportion of the time in each state.

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Answered: Q6 At all times, an urn contains N…

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″
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Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities P;,j: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N = 2. Then the states are 0, 1, 2 and the transition probability matrix is given by (i) 2 2 2 2 Find the proportion of the time in each state.
2. Suppose you are at a birthday party with n other people. How large must n be to ensure that the probability that you share your birthday with at least one other person is greater than 1/2?

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Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0