Q. 3 (Buttons). A monkey is presented with a red and a green button. The green button provides him with food, and the red button lets him use the bathroom. He repeatedly presses the buttons. Let Xk be the color of the kth button pressed by the monkey. Assume that {Xk} are independent and P{Xk = red} = 1 3 , P{Xk = green} = 2 3 . 1 a. Let Rn be the number of times the monkey pressed the red button after pressing n buttons in total, and let Gn be the number of times the monkey pressed the green button. What is the joint distribution of Rn and Gn? Are Rn and Gn independent? b. Let N be a Poisson random variable with mean a, independent of {Xk}. What is the joint distribution of RN and GN ? Are RN and GN independent?