(bh 7.70) in humans (and many other organisms), genes come in pairs. consider agene of interest, which comes in two types (alleles): typeaand typea. thegenotypeof a person for that gene is the types of the two genes in the pair: aa, aa, oraa(aaisequivalent toaa). according to the hardy-weinberg law, for a population in equilibriumthe frequencies ofaa, aa, aawill bep2,2p(1--p)2respectively, for somepwith0< p < 1.suppose that the hardy-weinberg law holds, and thatnpeople are drawnrandomly from the population, independently. letx1, x2, x3be the number of people inthe sample with genotypesaa, aa, aa, respectively.

(a) what is the joint pmf ofx1, x2, x3?

(b) what is the distribution of the number of people in the sample who have ana?

(c) what is the distribution of how many of the 2ngenes among the people area’s?

(d) now suppose thatpis unknown, and must be estimated using the observed datax1, x2, x3. themaximum likelihood estimator(mle) ofpis the value ofpfor which theobserved data are as likely as possible. find the mle ofp.

(e) now suppose thatpis unknown, and that our observations can’t distinguish betweenaaandaa. so for each person in the sample, we just know whether or not that person isanaa(in genetics terms, aaandaahave the samephenotype, and we only get to observethe phenotypes, not the genotypes). find the mle ofp.7.(bh 7.76) let (x, y) be bivariate normal withx∼ n(0, σ21) andy∼ n(0, σ22)marginally and with corr(x, y) =rho. find a constantcsuch thaty-cxis independentofx. hint: first findc(in terms ofrho, σ1, σ2) such thaty-cxandxare uncorrelated.3