Web hosting directory - 134 RANDOM NUMBERS 3.4.1 v 7. [,80] (A.
134 RANDOM NUMBERS 3.4.1 v 7. [,80] (A. J. Walker.) Suppose we have a bunch of cubes of k different colors, say nj cubes of color C, for 1 2 j 5 k, and we also have k boxes {Bl, . . . , Bk} each of which can hold exactly n cubes. Furthermore n1 + +. . + nk = kn, so the cubes will just fit in the boxes. Prove (constructively) that there is always a way to put the cubes into the boxes so that each box contains at most two different colors of cubes; in fact, there is a way to do it so that, whenever box Bj contains two colors, one of those colors is Cj. Show how to use this principle to compute the P and Y tables required in (3), given a probability distribution (~1, . . . , pk). 8. [ML51 Show that operation (3) could be changed to if U < PK then X + zK+l otherwise X t YK (i.e., using the original value of U instead of V) if this were more convenient, by suitably modifying PO, PI, . . , P&-l. 9. [HA&O] Why is the curve f(z) of Fig. 9 concave downward for 2 < 1, concave upward for x > l? b 10. [HM24] Explain how to calculate auxiliary constants Pj, Q3, Yj, Zj, Sj, Dj, Ej so that Algorithm M delivers answers with the correct distribution. b 11. [HM27] Prove that steps M7-M8 of Algorithm M generate a random variable with the appropriate tail of the normal distribution; i.e., the probability that X 5 x should be JI e-t 12dtl~ue- ./2dt, x 2 3. [Hint: Show that it is a special case of the rejection method, with g(t) = Ctept2/ for some C.] 12. [HMZI] (R. P. Brent.) Prove that the numbers oj defined in (23) satisfy the relation a;/2 -a;-,/2 < In 2 for all j 2 1. [Hint: If f(x) = ez2/ s, e–t2/2 dt, show that f(x) < f(y) for 0 I x < y.] 13. [HM.%] Given a set of n independent normal deviates, Xi, X2, . , X,, with mean 0 and variance 1, show how to find constants bj and az3, 1 5 j 2 i 5 n, so that if Yl = bl + al&l, Yz = bz + ~21×1 + azzX2, . . . , Yn = b, + ~1×1 + an2X2 +. . . + annXn, thenYi,Yz, . . . . Y, are dependent normally distributed variables, Yj has mean pj, and the Y s have a given covariance matrix (ci3). (The covariance, cij, of Yi and Yj is defined to be the average value of (Yz - pt)(Yj -pLj). In particular, Cjj is the variance of Yj, the square of its standard deviation. Not all matrices (cij) can be covariance matrices, and your construction is, of course, only supposed to work whenever a solution to the given conditions is possible.) 14. [MZ] If X is a random variable with continuous distribution F(x), and if c is a constant, what is the distribution of cX? 15. [HM.U] If Xl and X2 are independent random variables with the respective distributions Ft(x) and Fz(x), and with densities jr(x) = Fl (x), f2(x) = Fz (x), what are the distribution and density functions of the quantity X1 +X2?