Free web hosting with ftp - 3.4.1 ANSWERS TO EXERCISES 553 This approach applies
3.4.1 ANSWERS TO EXERCISES 553 This approach applies also to the binomial distribution, with 1 G(x) = U=- (l -U)n–2 dur(t + l)/r(z)r(t + 1 -x), s P since [G- (U)] is binomial with parameters (t, p) and G is approximately normal. [See also the alternative method proposed by Ahrens and Dieter in Computing (1980), to appear.] 23. Yes. The second method calculates Icos201, where 0 is uniformly distributed between 0 and 7rf 2. (Let U = r cos 0, V = r sin 0.) 25. #& = (.10101)2. In general, the binary representation is formed by using 1 for V and 0 for A, from left to right, then suffixing 1. This technique [cf. K. D. Tocher, J. Roy. Stat. Sot El6 (1954), 491 can lead to efficient generation of independent bits having a given probability p, and it can also be applied to the geometric and binomial distributions. 26. (a) True, c, Pr(N1 = k)Pr(Nz = 12 - k) = e-p1-@2(p1 + pz) /n!. (b) False, unless ~2 = 0; otherwise iV1 -N2 might be negative. 27. Let the binary representation of p be (&lb&.. .)2, and proceed according to the following rules: Bl. [Initialize.] Set m t t, N + 0, j + 1. (During this algorithm, m represents the number of simulated uniform deviates whose relation to p is still unknown, since they match p in their leading j-l bits; and N is the number of simulated deviates known to be less than p.) B2. [Look at next column of bits.] Generate a random integer M with the binomial distribution (m, a). (Now M represents the number of unknown deviates that fail to match b3.) Set m t m -M, and if b3 = 1 set N + N+M. B3. [Done?] If m = 0, or if the remaining bits (.b3+lb3+2.. .)z of p are all zero, the algorithm terminates. Otherwise, set j + j f 1 and return to step B2. 1 [When bJ = 1 for infinitely many j, the average number of iterations At satisfies A,, = 0; An=l+&~ ; Ak, for n 2 1. 0 k LettingA = zAnzn/n!, we haveA = e -l+A(~z)e /2. ThereforeA(z)e- = 1 - e- + A( $z)e-Z/2 = Ck,0(l-e-Z/2 ) = 1-e-Z-C,,1(-z)n/(7z!(2n -l)), and A,=l+C ; E =1+t V 0 n+l k>l in the notation of exercise 5.2.2.-48.1 28. Generate a random point (~1,. . . , yn) on the unit sphere, and let p = dm. Generate an independent uniform deviate U, and if pn+ U < K dm, output the point (yl/p, . . . , y,/p); otherwise start over. Here K2 = min{ (c aky~) + /(~ a:~:) 1 Cy&l}=a;--l f I nan >_ al, ((n + l)/(al f a7L))n+1(ulan/n)n otherwise.
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