Genetics A population is said to be in HardyWeinberg equilibrium, with respect to a

Chapter 5, Problem 9

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Genetics A population is said to be in HardyWeinberg equilibrium, with respect to a single gene with two alleles A and a, if the three genotypes AA, Aa, and aa have respective frequencies pAA = 2, pAa = 2(1), and paa = (1)2 for some [0, 1]. Suppose that we take a random sample of size n from a population. We can show that the probability of observing n1 individuals of type AA, n2 individuals of type Aa, and n3 individuals of type aa is given by n! n1! n2! n3! pn1 AA pn2 Aa pn3 aa where n! = n(n 1)(n 2) 3 2 1 (read n factorial). Here, n1 + n2 + n3 = n. This probability depends on . There is a method, called the , that can be used to estimate . The principle is simple: We find the value of that maximizes the probability of the observed data. Since the coefficient n! n1! n2! n3! does not depend on , we need only maximize L() = pn1 AA pn2 Aa pn3 aa (a) Suppose n1 = 8, n2 = 6, and n3 = 3. Compute L(). (b) Show that if L() is maximal for = (read theta hat), then ln L() is also maximal for = . (c) Use your result in (b) to find the value that maximizes L() for the data given in (a). The number is the maximum likelihood estimate.

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