# Download Characterisation of Probability Distributions by Janos Galambos, Samuel Kotz PDF

By Janos Galambos, Samuel Kotz

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**Extra info for Characterisation of Probability Distributions**

**Sample text**

The assump- function. This completes the proof. When g(y) is a given constant then we get back the negative exponential distribution. But since in most practical applications, a natural assumption is that 32 g(y) is strictly decreasing, (28) provides the population distribution for a wide variety of problems. ) is a given function. (29) In considering the equation E[h(X) BX~z] = g ( z ) , z ~ 0, P(X~0) = 1 , however, we h a v e t o b e c a r e f u l n o t t o r e p e a t thus getting trivialities (2) i n d i s g u i s e d b u t e q u i v a l e n t (see the discussion following (2a)).

Then for any A < 0, That is, for any finite number A, F(A) > 0 48 (21) P(XI: 2 < A, X2:2 - XI: 2 k ]A]) ~ 2P(XI

2. Let X 1 and X 2 be independent random variables with common dis- tribution function F(x). If XI: 2 and X2:2 - XI: 2 are independent, either discrete or F(x) = 1 - exp[-b(x-B)], then F(x) is x ~ B, where b > 0 and B are finite constants. 3, that there are discrete distributions XI: 2 and X2:2 - XI: 2 are independent. for which One additional class of discrete distributions is the degenerate one for which the above independence property still holds. It is, however, true that there are no other discrete distributions with this independence 50 property.