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By Robert V. Hogg, Stuart A. Klugman

Dedicated to the matter of becoming parametric chance distributions to information, this therapy uniquely unifies loss modeling in a single booklet. facts units used are concerning the assurance undefined, yet might be utilized to different distributions. Emphasis is at the distribution of unmarried losses concerning claims made opposed to numerous sorts of policies. contains 5 units of coverage information as examples.

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Nk and a common p is binomial with parameters nl + n2+ . . + nk and p (Exercise 3). 4. If X I , X 2 ,. . , X , are n mutually independent normal variables with means p l , p2,. . ,a:, . . , a;, respectively, then See Exercise 4. f. = = [ ( X - p ) ) / a I 2has (1 - 2t)-"2, provided 1 - 2t > 0, that is, t < i. This means that if XI, X 2 ,, . , X, are normally distributed as hypothesized in result (4),then [ ( X i- p i ) / a j ] 2is x2(1),i = 1, 2 , . . , n ; and by result (l), an important fact in statistics.

Rk degrees of freedom (Exercise 1). 2. The sum of k mutually independent Poisson random variables with respective means p l , p2, . . , pk is Poisson with mean p l + p2 + . . + pk (Exercise 2). 3. The sum of k mutually independent binomial random variables with respective parameters n l , n 2 , .. , nk and a common p is binomial with parameters nl + n2+ . . + nk and p (Exercise 3). 4. If X I , X 2 ,. . , X , are n mutually independent normal variables with means p l , p2,. . ,a:, . .

We call the associated distribution the transformed Pareto distribution or Burr distribution (Burr, 1942) and it has proved to be a useful one in modeling certain thicker tailed distributions. s. There is another important type of compound distribution that must be mentioned. f. f. technique. However, let us say that n, the number of claims, is the value of a random variable that has either a Poisson distribution with mean p or a negative binomial distribution with parameters r and p . f. g(x) of X is g(x)= T f n ( X ) F n=O n.

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