exercises for Mon Oct 12

1. On Mon Oct 5, we discussed Nils Exercises 14, 15, in particular touching on parametric CD inference for Cox type data, (t_i, x_i, \delta_i). We also went through aspects of the somewhat complicated CLP Exercise 4.3, "complicated" in the sense that its interpretation is not quite clear. The Natural Start of modelling the three trinomial probabilities as lambda pr1 + (1-lambda) pr2, where pr1 = (p_1^2, 2 p_1q_1, q_1^2) and pr2 = (p_2^2, 2 p_2q_1, q_2^2), *doesn't work*. Explain why.

2. Note that I've now placed the firetimerseksamen 2016 on the website.

3. Exercises for Mon Oct 12: First, Exercises 1, 3 from this firetimerseksamen set.

Then, consider the following Watership Down scenario. Rabbits have been living for fifty years on an island, and their alleles a, A are in good Hardy-Weinberg balance, with probabilities of the form prob = (p^2, 2pq, q^2) for aa, aA, AA, with known p = 0.25 and q = 0.75. Then there's an invasion of New Rabbits, with their own HW balance of alleles, with known p' = 0.40 and q' = 0.60. The two populations don't mix, though they look alike to the rabbitologists. Suppose they examine n = 1000 randomly sampled rabbits, with genetic testing, and find (X, Y, Z) = (89, 415, 496) in the three categories aa, aA, AA.

(a) Estimate the proportion \lambda of New Rabbits on the island, supplemented with a confidence curve.

(b) Suppose (p, q) = (0.25, 0.75) for the Indigenous Aboriginal Rabbits are known, but that (p', q') for the New Rabbits is not. Estimate both the proportion of New Rabbits and their (p', q').

(c) Explain why the problem of estimating each of \lambda, p, p' can't be solved based on (X, Y, Z) alone.