Identical twins are rare, but just how unlikely are they? With the help of the sociology department, you have a representative sample of twins to help you answer the question. The twins data gives the following observations (a twin refers to one pair of two people):

• m_i = number of identical male twins and f_i = number of identical female twins

• m_f = number of fraternal male twins and f_f = number of fraternal female twins

• b = number of fraternal opposite gender twins To model this data, we choose these distributions and parameters:

• Twins are identical with probability θ.

• Given identical twins, the twins are male with probability p.

• Given fraternal twins, the probability of male twins is q², probability of female twins is (1 − q)² and probability of oppsite gender twins is 2q(1 − q).

a. Write expressions for the likelihood of the data as a function of the parameters θ, p, and q for the observations m_i, f_i, m_f, f_f, b.

b. What are the maximum likelihood estimates for θ, p and q?