Download Analytic Function Methods in Probability Theory: Colloquium by B. Gyires PDF

By B. Gyires

ISBN-10: 0444853332

ISBN-13: 9780444853332

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Extra resources for Analytic Function Methods in Probability Theory: Colloquium Proceedings (Colloquia mathematica Societatis Janos Bolyai ; 21)

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2 (2ac -\)-- 2c c=a+b. The joint distribution of R and B is given by THEOREM 2. 9) 2 2 + L n=1 2 n(n-I) e- 2u (n-v) _ L n-v (0 -2u n n (n+l) n-tv n=1 < u < By putting 8 2 '" 2 -2u n - vu '" n e n=1 2 e -2u (n+v) o < 00 v 2 < I). has been determ- The marginal distribution of ined by GNEDENKO [5J. 9) one obtains 2 2 (2. 10) 1+2 L (1_4n 2 u 2 )e- 2u n n=1 or using theta function identities an equivalent expre- + ssion is given by (2. 9), one gets the u distribution of P(Q n lt 2 v'2"i ~ n= I 2 2 n -v (2.

2 in general, be improv- On the other hand, we saw that we have troubles with p's only, where the ball differentiability in those knocks against different dimensional (n-I ,n-2, ••• ,n) faces of the boundary of the In fact, easy to prove that A n (p) I- simplex. it is is piecewise analytic on 63 - (0,00) • 4. A (p(x» n (4. pn(x>{S - ~ n i=1 , An- I , 3=··· =A n- I ,n+ I where I =--; A =A lIn n-I, I n - I =nn/2/ r (I 2 +1), since is easily seen that follows then, points of the c S is an inner point of n sn and n S It n do not have obtuse angles.

2) = f(I), I f(-Z) = I satisfies the functional equation x f (x) + (I-x) f (-y-) f(y) + (I-y) f(-I- ) -y I-x for all (1. 3) is said to be an information I] if function (1. I) [0, defined on f (x,y)ED, where D {(x,y): 0 S x < I, 0 S Y < I, x+y S I}. The following result has been proved by Z. DAROCZY , and I. f KATAI [4J. Let be an information function. 4) =\ If then in [O,IJ 0 if xE(O,I) if x=O or x=1 is the Shannon information function. J. ACZEL and Z. 84): S Is the only nonnegative information function?

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