[36 points] Suppose X is a discrete random variable chosen from the uniform distribution over the numbers {1, 2, . . . ,10}. Based on the value that X takes (say X = :13), we dene a. second discrete random variable Y by choosing it uniformly at random between 1 and 3:. For example, if X = 3, then Y is a random variable chosen from the uniform distribution over the numbers {1, 2, 3}. If X = 5, then Y is a random variable chosen from the uniform distribution over the numbers {1, 2, 3, 4, 5}. (And if X = 1, Y must take the value of 1.) In this question, express any numerical answers as decimals with four signicant digits. (a) Are X and Y independent? Explain. (b) What is P (Y = 6)? (c) Suppose we observe Y = 6. Conditioned on this observation, what is the probability that X = 4? That is, what is P (X = 4|Y = 6)? (d) Suppose we observe Y = 6. Conditioned on this observation, what is the probability that X = 8? That is, what is P (X = 8|Y = 6)