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Question 3 25 marks This question is intended to assess your understanding of random walks on the line. You should be able to answer this question aer working through Part]! of Book 3. (a) Suggest an example, different from any mentioned in Part II or in Section 15 of Book 3, of a situation that could reasonably be modelled by a particle executing a simple random walk on the line with two absorbing barriers. Be sure to explain why a simple random walk might be a suitable model and what absorption at the barriers represents. (b) A particle executes an unrestricted random walk on the line starting at the origin. The ith step, 2;, has the following distribution: P(Zi=3)=p, P(Zi=2)=q=1P- (i) Find the mean and variance of Z;, and hence nd the mean and variance of X", the position of the particle after 11 steps. (ii) Find the probability distribution of X", and state the range of X\". Explain all the steps in your derivation of the distribution (iii) Calculate the following probabilities in terms of p and q: P(X5 = 2), P(X2 = 3) (iv) Calculate the probabilities uo, us, we in terms of p and q, and hence find the probability that the particle returns to the origin for the rst time after 10 steps. [4] [5] [6] [5] [5]