Question 2 (10 points). The ability to roll one's tongue is often thought of as a trait that can be inherited from parents. Let MR,FR, and CR represent the random variables that the mother, father, and child can roll their tongues respectively. Let MG,FG, and CG represent the random variables of mother gene, father gene, and child gene respectively. All the variables can take on either one of two possible Boolean outcomes: t and f for true and false. It is a usually considered that genes can cause the person to roll their tongue. We model the possible relationships between the variables using 3 different Bayesian networks as shown below. (i) (ii) (a) Which of the three networks claim that P(MG,FG,CG)=P(MG)P(FG)P(CG) ? Explain your answer. (2 points) (b) Consider following hypothesis for the inheritance of tongue rolling: "Tongue rolling is caused by the gene, which the child inherits from his/her parents". We assume that this inheritance of genes is the only way that the parents can influence the presence of the trait in the child. Which of the three networks make independence claims that are consistent with the hypothesis? Explain your answer. Your answer may contain zero, one, or more than one network. (4 points) (c) The CPT for the CG node in network (i), in terms of x and y, is given in the table below. In network (i), suppose that P(FG=f)=P(MG=f)=z, and derive an expression for P(CG=t) in terms of x, y and z only, by conditioning on its parent nodes. (2 points) (d) Under certain conditions that correspond to genetic equilibrium, we can expect the distribution of genes to be the same across generations, i.e., P(CG=t)=P(FG= t)=P(MG=t)=1z. Let z=0.4. You are now tasked to use this information and calculate the value of x. In addition, what conclusion can you draw about the child gene (CG), when you are given that MG=FG ? (2 points)