Bulletproof web design - 56 RANDOM NUMBERS 3.3.1 The more precise asymptotic
56 RANDOM NUMBERS 3.3.1 The more precise asymptotic formulas in Table 2 follow from results obtained by D. A. Darling [Theory of Prob. and Appl. 5 (1960) 356-3611, who proved among other things that K,+ 5 s with probability 1 - CZSZ(l -$s/fi + 0(1/n)) (28) for all fixed s 2 0. EXERCISES 1. [OO] What line of the chi-square table should be used to check whether or not the value V = 7& of Eq. (5) is improbably high? 2. [Zoo] If two dice are loaded so that, on one die, the value 1 will turn up exactly twice as often as any of the other values, and the other die is similarly biased towards 6, compute the probability ps that a total of exactly s will appear on the two dice, for 2 2 s 2 12. b 3. [23] Some dice that were loaded as described in the previous exercise were rolled 144 times, and the following values were observed: value of s = 2 3 4 5 6 7 8 9 10 11 12 observed number, Y, = 2 6 10 16 18 32 20 13 16 9 2 Apply the chi-square test to these values, using the probabilities in (l), pretending it is not known that the dice are in fact faulty. Does the chi-square test detect the bad dice? If not, explain why not. b 4. [z?] The author actually obtained the data in experiment 1 of (9) by simulating dice in which one was normal, the other was loaded so that it always turned up 1 or 6. (The latter two possibilities were equally probable.) Compute the probabilities that replace (1) in this case, and by using a chi-square test decide if the results of that experiment are consistent with the dice being loaded in this way. 5. [,B] Let F(z) be the uniform distribution, Fig. 3(b). Find K& and KG for the following 20 observations: 0.414, 0.732, 0.236, 0.162, 0.259, 0.442, 0.189, 0.693, 0.098, 0.302, 0.442, 0.434, 0.141, 0.017, 0.318, 0.869, 0.772, 0.678, 0.354, 0.718, and state whether these observations are significantly different from expected behavior with respect to either of these two tests. 6. [A&V] Consider Fn(z), as given in Eq. (lo), for fixed 2. What is the probability that Fn(z) = s/n, given an integer s? What is the mean value of Fn(z)? What is the standard deviation? 7. [MU] Show that Kz and K; can never be negative. What is the largest possible value K,+ can be?
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