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1, A life insurance company has to date only sold a small range of simple product.It is considering diversifying it's product range to offer a more complete range of products

Explain the benefits such a diversifications might provide for the insurance company.

An author is about to start writing a book that will contain 20 chapters. The author plans to write a new chapter each week. However, when he reviews his work at the end of each week, there is a probability of 0.25 (which is independent of the current state of the book) that he will not be happy with one of the chapters he has written. In this case, he will spend the following week rewriting that particular chapter instead of embarking on a new one. He may decide to rewrite any one chapter, including a new one he has just finished or one that he has previously rewritten. Let X, denote the number of chapters that the author is happy with at the end of week k , and define Xo = 0. (i) Explain why Xx can be modelled as a Markov chain. [2] (ii) Calculate the probability that the author will complete the book in exactly 25 weeks. [2] (iii) Calculate the expected number of weeks it will take the author to complete the book. [2] [Total 6]