Problem 1 A monkey climbs a ladder with n bars, so they jump to the next bar at rate A but fall at the bottom of the ladder independently at rate . Once at the top, the monkey can only fall with rate . a. Consider the state space S = {0,1,..., n} with the bottom to be at 0 and the top of the ladder at n. For n 4 write the matrix Q (with parameters qi,j) of the CTMC associated with the position of the monkey. b. In the general case, find the fraction of time the monkey spends at the bottom of the ladder, as a function of a /(\+). =