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By Luke Y.L.

ISBN-10: 0124599400

ISBN-13: 9780124599406

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K 1/ C 2 n u2n ; kD1 where u2n 1= p n, and conclude that r E min. n ; 2n/ 4 n ; n ! 2. Let n D minf1 Ä k Ä nW Sk D 1g, with the understanding that n D 1 if Sk < 1, for all 1 Ä k Ä n. What is the limit of E min. n ; n/ as n ! p ¤ q/ Bernoulli walk? Hint. p q/ 1 ; p > q; E min. n ; n/ ! 3. mod 2/, in all other cases ; where k D 0; ˙1; : : : ; ˙n. n r/=2c 2 n ; 0ÄkÄn for r D 0; 1; : : : ; n. 4. Let 1 ; : : : ; 2n be independent Bernoulli random variables with Pf k D 1g D Pf k D 1g D 1=2, k Ä 2n. S2 ; S4 ; : : : ; S2n /, where we set g2n D 0 if no such moment exists.

Based on a probabilistic argument, assuming that the number n of all missprints in the book is quite large, one can suppose that an and bn are reasonably close to the probabilities pa and pb for a misprint to be detected, respectively, by corrector A and corrector B. 1. Give an example of two random variables, independent and yet the relation E. 8, (22)]). 2. The conditional variance of the random variable the partition D is defined as the random variable D. j D/ D EŒ. 43 with respect to E. j D//2 jD : Prove that the variance of satisfies the relation: D D ED.

8. 2. x/ is some non-negative function which is even and is also non-decreasing for positive x. /j Ä C , C > 0, verify the following estimate: Pfj j Ef . x/ D x 2 one must have E 2 "2 C2 Â Ä Pfj E j D Ä 2 " "g à : Hint. Use the following relation Ef . 3. Let 1 ; : : : ; with D i Ä C . "/ : be any sequence of independent random variables n E. 4. Suppose that 1 ; : : : ; n are independent Bernoulli random variables with Pf i D 1g D p > 0 and Pf i D 1g D 1 p. 2p ˇ ˇ 1/ˇ o " Ä 2e a"2 n ; where Sn D 1 C : : : C n and " > 0.

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Algorithms for computations of mathematical functions by Luke Y.L.

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