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HardyWeinberg equilibrium problem if the three genotypes AA, Aa, and aa have respective frequencies pAA =θ^2, pAa = 2θ(1 − θ), and paa = (1 − θ)^2, n! / (divide) n1! n2! n3! PAA^n1 PAa^n2 Paa^n3 where n = n1 + n2 + n3. This probablity depends on θ. There is a method called the maximum likelihood method, that can be used to estimate θ. The principle is simple: We find the value of θ that maximizes the probability of the observed data. Since the coefficient n! / (divide) n1! n2! n3! does not depend on θ, we need only maximize L(θ) = PAA^n1 PAa^n2 Paa^n3 (a) Show that if L(θ) is maximal for θ = ˆθ then ln(L(θ)) is maximal for θ = ˆθ. (Note that L is strictly positive and twice differentiable.) (b) Use the result in (a) to find the value ˆθ that maximizes L(θ)
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