Hi! I need help with the following questions:

2. A man is trapped in a room at the center of a maze. The room has three exits. Exit 1 leads outside the maze after 3 minutes, on average. Exit 2 will bring him back to the same room after 5 minutes. Exit 3 will bring him back to the same room after 7 minutes. Assume that every time he makes a choice, he is equally likely to choose any exit. What is the expected time taken by him to leave the maze? Hint: Let X = time taken by the man to leave the maze from this room. Let Y = exit he chooses rst. So Y belongs in {1, 2, 3}. Calculate the conditional expectation of time taken to leave the maze given that he chose each of the exits. Then use these conditional expectations to calculate the expectation of time taken to leave the maze. 3. Consider two random variable X and Y with joint PMF give in the table below. Y=2 Y=4 =5 x=1 k/3 k/6 k/6 x=2 2k/3 k/s r x=3 k k/z k/3 7 a) Find the value of k so that this is a valid PMF. Show your work. b) Re-write the table with the joint probabilities using the value of k that you found in (a). c) Find the marginal PlVEFs of X and Y. Show your work. d) Find P(X S 2, Y S 4) e) Find P(Y =2 l X=1). Show your work. t) Are X and Y independent? Justify with calculations. 4. If X and Y are discrete random variables with joint PMF my P(X1)(x~_V)=C ' r xx'" ,forx20,1, 2, andyZO, 1, 2, and zero otherwise. a) Find the constant e. b) Find the marginal PMFS of X and Y. Identify their distribution along with their parameters. o) Are X and Y independent? Why/why not