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quiz_05 - Math 213 Quiz 5 Name Solution 1 Three...

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Math 213 — Quiz 5 Name Solution 1. Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy–Weinberg law states that the proportion of individuals in a population who carry two diﬀerent alleles is P = 2 pq + 2 qr + 2 rp where p , q , and r are the proportions of A, B, and O in the population. Use the fact that p + q + r = 1 (and p,q,r 0) to show that P is at most 2 / 3. First we use r = 1 - p - q to reduce dimension by one. We want to ﬁnd the maximum of f ( p,q ) = P ( p,q, 1 - p - q ) = 2 pq +2 q (1 - p - q )+2 p (1 - p - q ) = 2 p +2 q - 2 pq - 2 p 2 - 2 q 2 over the triangular region D = { ( p,q ) : p 0 ,q 0 ,p + q 1 } . (The restriction p + q 1 is equivalent to r 0.) The gradient of f is f = h 2 - 4 p - 2 q, 2 - 2 p - 4 q i . So the only critical point of f in the interior of D is when p = q = 1 / 3. This corresponds to the distribution ( p,q,r ) = (1 / 3 , 1 / 3 , 1 / 3) and P (1 / 3
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