6.2.19 (Hardy-Weinberg law) The Hardy-Weinberg law in genetics says that the pro- portions of genotypes AA, Aa, and aa are 02, 20 (1 - 0), and (1 - 0), respectively, where 0 E [0, 1]. Suppose that in a sample of n from the population (small relative to the size of the population), we observe x1 individuals of type AA, x2 individuals of type Aa, and x3 individuals of type aa. (a) Now find Wald and score intervals for 0. (b) The probability for the AA genotype is 02. Use the delta method to get an asymptotic variance for the MLE of 02 and then use that to construct a confidence interval for 02. Given: The distribution is multinomial(02, 20(1 - 0), (1 -0)2). ((0) = (2X1+ X2) log(0) + (2X3 + X2) log(1 -0) + log(n!) + X2log(2) - log(X1!X2!X3!). The score equation is S(0) = 2X14X2 _ 2X3+X2 = 0, which leads to the MLE 0 = 1-0 2X1+X2 2(X1+X2+X3) 2n 2X1+X2. Note that S(0) is decreasing