By Joel Amato, Steve Mack

ISBN-10: 015364057X

ISBN-13: 9780153640575

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189) 12. More generally, Pr n Ej = Pr(E1 ) Pr(E2 |E1 ) Pr(E3 |E1 ∩ E2 ) · · · Pr En j =1 n−1 Ej . 190) 13. 189) becomes Pr(E1 ∩ E2 ) = Pr(E1 ) Pr(E2 ). 191) 14. We say that n events are mutually independent if, for every subset {Ej1 , Ej2 , . . , Ejk }, k ≤ n, k Pr(Ej1 ∩ Ej2 ∩ · · · ∩ Ejk ) = Pr(Eji ). 192) i=1 j −1 15. 190) simpliﬁes to n Pr(E1 ∩ E2 ∩ · · · ∩ En ) = Pr(Ej ). 193) 41 PROBABILITY AND STATISTICAL PRELIMINARIES 16. The theorem of total probability states that, if E1 , E2 , .

37) is called the (s + 2)-gamma function. Extensive tables of the digamma, trigamma, tetragamma, pentagamma, and hexagamma functions are contained in Davis (1933, 1935). Shorter tables are in Abramowitz and Stegun (1965). 19) for the gamma function yields the following recurrence formulas for the psi function: ψ(x + 1) = ψ(x) + x −1 and n ψ(x + n) = ψ(x) + (x + j − 1)−1 , n = 1, 2, 3, . . 40) j =0 and ψ(mx) = ln(m) + 1 m m−1 ψ x+ j =0 j , m m = 1, 2, 3, . . 5772156649 . ). 5), provided that x ≥ 2.

The Mellin transform [R. H. 158) 0 f (x) = 1 2πi c+i∞ c−i∞ x −s H (s) ds. 158) gives the (s − 1)th moment about the origin of a distribution on the nonnegative real line. Springer (1979) has demonstrated the key role of the Mellin transform and its inverse in the derivation of distributions of products, quotients, and other algebraic functions of independent random variables. For a comprehensive coverage of these and other types of integral transforms, see Erd´elyi et al. (1954, Vols. 1 and 2).

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