Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at A, B, C, and D. One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit. Once at a friend’s house, he will either return home or else proceed to one of the two adjacent houses (such as 0, A, or C when at B), with each of the three possibilities having probability . In this way, Alvie continues to visit friends until he returns home. a. Let X = the number of times that Alvie visits a friend. Derive the pmf of X. b. ?Let Y= the number of straight-line segments that Alvie traverses (including those leading to and from 0). What is the pmf of Y? c. ?Suppose that female friends live at A and C and male friends at B and D. If Z = the number f visits to female friends, what is the pmf of Z?

Answer: Step 1 of 1 Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at A, B, C, and D. One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit. Once at a friend’s house, he will either return home or else proceed to one of the two adjacent houses (such as 0, A, or C when at B), with each of the three possibilities having probability. a) Alvie visits 1 friend = (1/3) Alvie visits 2 friends = (2/3)(1/3) Alvie visits 3 friends = (2/3) (1/3) 3 Alvie visits 4 friends = (2/3) (1/3) Now, Alvie visits x friends = (2/3) (1/3) Where X is the number of number of times Alvie visits a friend. Thus, b) Here, Alvie will travel on x+1 straight line...