Stats: mle with two parameters

[bennyska]

Homework Statement

in a genetics situation, we have two variables, x1 and x2, such that both x1 and x2 >0, and x1+x2<1.
we have:
p1 = x1²
p2 = x2²
p3 = (1-x1-x2)²
p4 = 2x1x2
p5 = 2x1(1-x1-x2)
p6 = 2x2(1-x1-x2)

find the mles for x1 and x2.

Homework Equations

the answer (from the book): x1(hat) = (2N1 + N4 + N5)/2n, where n = sum (Ni). x2(hat) is similar.

The Attempt at a Solution

doing the usual mle stuff, i have sum(Ni*ln(pi)), take the derivative, set to zero, and solve for my parameter. when i do this work, i get an answer similar (but incorrect) to that of the book, with a little problem: in order to get n, i need all the Nis. when i take the derivative with respect to x1, there is one term (p2) that doesn't have x1 in it, so it drops out, and i lose that term (N2) that i think i need. i can't define x1 by x2 (i think, since x1 + x2 < 1 doesn't tell me too much). my teacher assigned this one to us, without having done the problem herself, and then she found herself unable to do it. we figure there's some little trick involved, but I'm not seeing it. also, i haven't really done mles with more than one variable, so maybe that's what I'm missing. any help would be greatly appreciated.

 

[Stephen Tashi]

Homework Statement

in a genetics situation, we have two variables, x1 and x2, such that both x1 and x2 >0, and x1+x2<1.
we have:
p1 = x1²
p2 = x2²
p3 = (1-x1-x2)²
p4 = 2x1x2
p5 = 2x1(1-x1-x2)
p6 = 2x2(1-x1-x2)

.

We have them, but what are they? Are the p's functions?

 

[bennyska]

sorry for my vagueness. the actual problem is something like this: a certain gene has 3 alleles, resulting in six possible genotypes. each p is the probability of being a certain genotype. each parent has an x1 or x2 probability of passing that allele to the child.