By David C. M. Dickson
ISBN-10: 0521846404
ISBN-13: 9780521846400
In line with the author's event of training final-year actuarial scholars in Britain and Australia, and appropriate for a primary path in coverage hazard concept, this booklet specializes in the 2 significant parts of danger conception - mixture claims distributions and break conception. For combination claims distributions, certain descriptions are given of recursive suggestions that may be utilized in the person and collective chance types. For the collective version, diversified periods of counting distribution are mentioned, and recursion schemes for likelihood features and moments provided. For the person version, the 3 most ordinarily utilized options are mentioned and illustrated. Care has been taken to make the e-book obtainable to readers who've a pretty good figuring out of the fundamental instruments of likelihood conception. a number of labored examples are integrated within the textual content and every bankruptcy concludes with workouts, that have solutions within the ebook and whole suggestions on hand for teachers from www.cambridge.org/9780521846400.
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Extra resources for Insurance Risk and Ruin (International Series on Actuarial Science)
Sample text
Use the fact that the integral of this density function over (0, ∞) equals 1 to find the first three moments of a Pa(α, λ) distribution, where α > 3. 5. The random variable X has a Pa(α, λ) distribution. Let M be a positive constant. Show that E[min(X, M)] = λ 1− α−1 λ λ+M α−1 . 6. 4 to show that when X ∼ N (µ, σ 2 ), M X (t) = exp µt + 12 σ 2 t 2 . Verify that E[X ] = µ and V [X ] = σ 2 by differentiating this moment generating function. 7. Let the random variable X have distribution function F given by for x < 20 0 F(x) = (x + 20)/80 for 20 ≤ x < 40.
Define M S to be the moment generating function of Sn and define M X to be the moment generating function of X 1 . Then M S (t) = E et Sn = E et(X 1 +X 2 +···+X n ) . Using independence, it follows that M S (t) = E et X 1 E et X 2 · · · E et X n , and as the X i s are identically distributed, M S (t) = M X (t)n . Hence, if we can identify M X (t)n as the moment generating function of a distribution, we know the distribution of Sn by the uniqueness property of moment generating functions. 10 Let X 1 have a Poisson distribution with parameter λ.
Properties of the risk adjusted premium principle are discussed at length by Wang (1995). 5 Exercises 1. A premium principle is said to be sub-additive if for two risks X 1 and X 2 (which may be dependent), X 1 +X 2 ≤ X 1 + X 2 . Under what conditions is the variance principle sub-additive? 2. The mean value principle states that the premium, X , for a risk X is given by X = v −1 (E[v(X )]) where v is a function such that v (x) > 0 and v (x) ≥ 0 for x > 0. (a) Calculate X when v(x) = x 2 and X ∼ γ (2, 2).