Web hosting comparison - 58 RANDOM NUMBERS 3.3.1 b 16. [MM%] Generalize
58 RANDOM NUMBERS 3.3.1 b 16. [MM%] Generalize Theorem 1.2.11.3A to find the value of 7(x + 1, x + .zdz + Y)lUX + l), for large result to z and fixed y, z. find the approximate Disregard solution, terms t, to of the answer the equation that are 0(1/x). Use this Y($ po) =P, for large v and fixed p, thereby accounting for the asymptotic formulas indicated in Table 1. [Hint: See exercise 1.2.11.3-8.1 17. [HM96] Let t be a fixed real number. For 0 2 k 5 n, let by convention, let P&x) = 1. Prove the following relations: a)P~k(x)=~i+ dX~~~ldX,i …/ : dXk+l~zktldl* …lz2dxl. b) P,~(z) = (x + t) /n! -(x + t) - /(n -I)!. (k -t)k c) Pnk(x) -Pn(k-l)(X) = ~ P(+-k)o(x -k), if 1 5 k 5 n. k! d) Obtain a general formula for P&(x), and apply it to the evaluation of Eq. (24). 18. [A4%7] Give a simple reason why K; has the same probability distribution as K,+. 19. [HM@ Develop tests, analogous to the Kolomogrov-Smirnov test, for use with multivariate distributions F(xi, . , x7) = probability that (Xl < x1, . . . , X, 5 x7). (Such procedures could be used, for example, in place of the serial test in the next section.) 20. [HM41] Deduce further terms of the asymptotic behavior of the KS distribution, extending (28). 21. [M40] Although the text states that the KS test should be applied only when F(x) is a continuous distribution function, it is, of course, possible to try to compute K,f and K; even when the distribution has jumps. Analyze the probable behavior of K,f and K, for various discontinuous distributions F(x). Compare the effectiveness of the resulting statistical test with the chi-square test on several samples of random numbers. 22. [HM46] Investigate the improved KS test suggested in the answer to exercise 6. 23. [M,% .2] (T. Gonzalez, S. Sahni, and W. R. Franta.) (a) Suppose that the maxi- mum value in formula (13) for the KS statistic K$ occurs at a given index j where ]nF(Xj)] = k. Prove that F(Xy) = rnaxi~i~~{ F(Xi) 1 [nF(X,)] = k }. (b) Design an algorithm that calculates Kz and K; in O(n) steps (without sorting).
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