n0 + 1 r1 f1k s1k − − r11 ,...,r1k−1 ,s1k j=1 jr1j f1k ≤n1 −k s1k ≥1 n0 + 1 and this reduces to the formula in the theorem because the final two terms can be combined, and then can be summed over only runs that satisfy Lk−1 jr1j < n1 k and combined with the first − term (because f1k s1k = 0 if Lk−1 jr1j = n1 k).51 − − While the first term has the closed form representation L C1k = (n1 this does not appear to 51 s1k f1k −s1k −1)/(n−1), n0 +1 f1k k k −1 be so for the other terms. E[P1k = n1 ] cannot have a closed form representation in any case, as the term U1k |N1 does not allow one.
n0 + 1 − r1 f1k − s1k r11 ,...,r1k−1 ,s1k j=1 jr1j ≤n1 −k s1k ≥1 n0 + 1 f1k and this reduces to the formula in the theorem because the final two terms can be combined, and then can be summed over only runs that satisfy Lk−1 jr1j < n1 − k and combined with the first term (because f1k − s1k = 0 if Lk−1 jr1j = n1 − k).51 51 While the first term has the closed form representation L C1k s1k f1k −s1k = (n1 −1)/(n−1), this does not appear to n0 +1 f1k −1 k k be so for the other terms. E[P1k |N1 = n1 ] cannot have a closed form representation in any case, as the term U1k does not allow one.
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