By Dickson D.C.M.

ISBN-10: 0521846404

ISBN-13: 9780521846400

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1905. 7 Notes and references Further details of the distributions discussed in this chapter, including a discussion of how to fit parameters to these distributions, can be found in Hogg and Klugman (1984). See also Klugman et al. (1998). 3 was derived by De Pril (1985), and a very elegant proof of the result can be found in his paper. 8 Exercises 1. A random variable X has a logarithmic distribution with parameter θ , where 0 < θ < 1, if its probability function is Pr(X = x) = −1 θx log(1 − θ) x for x = 1, 2, 3, .

1 × 10 4 , 2000 2 ). Which of these investments does the investor prefer? 002). 002W } exp {−14} . Hence, the investor prefers Investment 1 as E [u(W + X 1 )] is greater than E [u(W + X 2 )]. Note that the expected utility criterion may lead to an outcome that is inconsistent with other criteria. This should not be surprising, as different criteria will, in general, lead to different decisions. 1 above, the investor did not choose the investment which gave the greater expected net gain. We end this section by remarking that if a utility function v is defined in terms of a utility function u by v(x) = au(x) + b for constants a and b, with a > 0, then decisions made under the expected utility criterion will be the same under v as under u since, for example, E [v(W + X 1 )] > E [v(W + X 2 )] if and only if a E [u(W + X 1 )] + b > a E [u(W + X 2 )] + b.

Where m is a positive integer. n X i and define f j = Pr(X 1 = j) for Let Sn = i=1 j = m, m + 1, m + 2, . . and g j = Pr(Sn = j) for j = mn, mn + 1, mn + 2, . . Show that gmn = f mn and for r = mn + 1, mn + 2, mn + 3, . . gr = 1 fm r −mn j=1 (n + 1) j −1 r − mn f j+m gr − j . 1 Introduction Utility theory is a subject which has many applications, particularly in economics. However, in this chapter we consider utility theory from an insurance perspective only. We start with a general discussion of utility, then introduce decision making, which is the key application of utility theory.

### Insurance risk and ruin by Dickson D.C.M.

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