New PDF release: Characterisation of Probability Distributions

By Janos Galambos, Samuel Kotz

ISBN-10: 3540089330

ISBN-13: 9783540089339

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Sample text

C n be c o n s t a n t s s a t i s f y i n g c k ~ 0, and c k + C k + l + . . + c n = O. + CnXn: n a r e i n d e p e n d e n t i f , and o n l y i f , F(x) = 1 - e x p [ - b ( x - B ) ] , x ~ B, where b > 0 and B a r e finite constants. P. Basu (1965). S. B. Kemperman (1971). They actually determine all distributions F and G under which min(Xl,X2) and IXI-X2] are independent, where X 1 and X 2 are independent random variables with distribution functions F and G, respectively. B. Crawford (1966) found that necessarily F and G are either both ex- 51 ponential or both geometric (see Chapter 6 for further details of the results by Ferguson, Crawford and Kemperman).

This constant value is evidently zero since lim R(u) = 0 as u ÷ +~. I ~ = l-F(u) e-y That is, for all y > 0 and u > 0. Letting u ÷ 0 we get F(y) = 1-e -y , y > 0, which was to be proved. Let us turn to question (ii). There are only a few solutions to this problem under rather restrictive assumptions. Some of these solutions, however, are very valuable in engineering applications. Namely, several failure models can be approx- imated by a model in which the components are independent (but not identically distributed).

Is a given number, then this e q u a t i o n becomes a c h a r a c t e r i s t i c property 31 of Fc(X ) with c = I/k. follows. That is, what we can conclude from this discussion is as If we assume that E(XIX_>z) is of a special form (z + constant), then sev- eral distributions can be obtained for X (Fc(X) with arbitrary c), but if we specify the unknowns as in the above equation (we give a meaning to the constant) then we arrive at a characterization theorem. With this general formulation we shall soon see that the linearity of E(XIX>-z) is not important.

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Characterisation of Probability Distributions by Janos Galambos, Samuel Kotz

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