3.3.2 EMPIRICAL TESTS 67 are (~+q+l)~~q)-(Pll~~l)-(p+~+l)+l (16) ways to arrange them in the order (15), as shown in exercise 13; and there are (n -p -q -l)! ways to arrange the remaining elements. Thus there are 1 1 times 16 ways in all, and dividing by n! we get the (p+:+&-P-q-1. ( 1 desired formula. From relations (14) a rather lengthy calculation leads to mean(R ,) = mean(Z,, + . . . + Z,,) = (n + l)Pl(P + 111 - (P - 1)/P!, l n, (18) where t = max(p, q), s = p + q, and ^, j(p, q, n) = (n + 1) s(l -pq) + pq 1Y -L)+(Y) (lg) ( (P + lY(q + (s + I)! , (s2 -s - 2)pq -s2 - p2q2 + 1 t. (P + lY(q + l)! . This expression for the covariance is unfortunately quite complicated, but it is necessary for a successful run test as described above. From these formulas it is easy to compute mean = mean(R ,) -mean(R ,+,), covar(Rp, Rb) = covar(RL, Rb) -covar(R ,+, , R ,), (20) covar(Rp, R4) = covar(R,, R ,) -covar(Rp, Rb+,). In Annals Math. Stat. 15 (1944), 163-165, J. Wolfowitz proved that the quantities Rl, R2, . . . , RtF1, R: become normally distributed as n –+ 00, subject to the mean and covariance expressed above; this implies that the following test for runs is valid: Given a sequence of n random numbers, compute the number of runs R, of length p for 1 2 p < t, and also the number of runs Ri of length t or more. Let &I = RI -mean( . . . . &t-l = Rt-1 -mean(Rt-I), &t = R$ -mean( (21)
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