Question 1 (30 points - 3 points each - Q1-10) A binary trait is classified on the basis of a pair of genes. Let d and r represent the dominant and recessive genes which determine the outward appearance of the trait. A person with dd, dr, or rd will show the dominant trait while a person with rr will show the recessive trait. Assume a child receives one gene independently from each parent with P(d) = 2/5 . 1) Let Y = outward appearance of the child (dominant appearance or recessive appearance). What is the distribution of Y ? 2) In a family of six children, each child independently shows either the outward appearance of the dominant or recessive trait. What is the probability that the first three children born in the family show the dominant trait and the last three show the recessive trait? 1 3) How many possible arrangements are there where we could have three children showing the dominant trait and three children showing the recessive trait? What is the probability of each individual arrangement? 4) What is the probability that we would obtain three children who show the dominant trait and three who show the recessive trait? 5) Let X be the number of children who show the dominant trait. What are the possible values of X? Determine whether X satisfies the conditions for a Binomial random variable. If it does, what are the parameters of the Binomial distribution for X? 6) Find the probability that more children in the family will show the recessive trait than the dominant one. 7) What is the expected value and variance of X? How does the binomial probability above compare to its normal approximation? 8) Suppose we wish to estimate the unknown probability of showing the dominant trait. We observe a family of N = 8 children and find that 3 show the dominant trait. Explain why = 3/8 would be the best choice of point estimator based on this observation. 9) Suppose now that we are interested in knowing the distribution of the genotype rather than the phenotype (outward appearance). Children are classified on the basis of whether they are purely dominant (dd), hybrid (rd or dr), or purely recessive (rr). Let X1,X2,X3 be the number of each genotype in a family of 6 with P(d) = 2/5 as before. What is the joint distribution of X1,X2,X3? 10) Find the probability of one pure dominant, two hybrids, and three recessives in the family of six. 11 - Extra credit - 10 points) In problem 4 above, you calculated the Binomial probability of three dominant and three recessive. Calculate that Binomial probability as the sum of four mutually exclusive multinomial probabilities.