4. (Total: 15 pit-r} Energ,r harvesting technolog,r allows for storage of energy derived from external sources {e.g.. solar power. thermal energy. ...] into batteries. onsider a battery that is used for completing tasks. and is charged by harvesting energy From the environment. The battery performs as follows: I At each unit of time. a task. is requested with probability 9 and the battery consumes 1 Joule in order to complete the requested task. If the battery is empty. no task can be done._ even if it is requested. I At each unit of time. 1 Joule of energy is harvested and stored in the battery with probability 19. independently from whether a task is requested or not. The battery has a capacity of 2' Joules. so it cannot store more than '2 Joules. even if it harvests energy. The process .171.th . . . represents the total energy stored in the battery {in Joules} as a function of time. This process can be represented as a Markov chain with states [t], l._ 2}. {a} {.5 jets} El'raur the state transition diagram of this Markov chain. {b} {2 jets} Find the probability transition matrix P. {c} {d guts} Suppose F{X1=1]= .fl_.P{X. = '2] = [1.7. i. 1What is the probability that the battery will be full at time n = 2. that is PEX-g = 2]? ii. 1|What is the probability that the battery will be empty at time n = 2'? iii. Find F{X;; = 1}. {d} {2 pm} Find HIKE-+2 = lXi = 2}. that is the probability that the battery goes from full to empty in two time slots