# Applied Statistics and the SAS Programming Language by Jfry KSmit

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By Jfry KSmit

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Simulans Esterase-C locus [n = 308] 91, 76, 70, 57, 12, 1, 1 It is clear that these data come from diﬀerent distributions. Of the ﬁrst set, Sewall Wright (1978, p303) argued that . . the observations do not agree at all with the equal frequencies expected for neutral alleles in enormously large populations. This raises the question of what shape these distributions should have under a neutral model. The answer to this was given by Ewens (1972). Because the labels are irrelevant, a sample of genes can be broken down into a set of alleles that occurred just once in the sample, another collection that occurred twice, and so on.

Bj , 0, . . , 0), and we assume this in the remainder of this section. We deﬁne ei = (0, 0, . . , 0, 1, 0, . . , 0), the ith unit vector. 1) with q(e1 ) = 1. Suppose then that the conﬁguration is c. Looking at the history of the sample, we will either ﬁnd a mutation or we will be able to 36 Simon Tavar´e trace two individuals back to a common ancestor. The ﬁrst event occurs with probability nθ/2 θ = , nθ/2 + n(n − 1)/2 θ+n−1 and results in the conﬁguration c if the conﬁguration just before the mutation was b, where (i) b = c, and mutation occurred to one of the c1 singleton lines (probability c1 /n); (ii) b = c − 2e1 + e2 , and a mutation occurred to an individual in the 2-class (probability 2(c2 + 1)/n); (iii) b = c − e1 − ej−1 + ej and the mutation occurred to an individual in a j-class, producing a singleton mutant and a new (j − 1)-class (probability j(cj + 1)/n).

Watterson (1974) noted that if Z1 , Z2 , . . are independent Poisson random variables with EZj = θ/j, then n L(C1 (n), C2 (n), . . , Cn (n)) = L Z1 , Z2 , . . ’ The ESF typically has a very skewed distribution, assigning most mass to conﬁgurations with several alleles represented a few times. In particular, the distribution is far from ‘ﬂat’; recall Wright’s observation cited in the introduction of this section. In the remainder of the section, we will explore some of the properties of the ESF.