Question 2. [12 marks]. A factory has produced m televisions which are for sale. Television i works for a random time Xi, where the Xi are independent and identically distributed with common distribution Exponential () where >0 is the rate parameter. Let N(t) denote the number of televisions that have failed by time t. (a) Calculate the probability P[N(t)=r], where r={0,1,,m}. (b) Let 1 denote the (random) time to the first failure. What is the distribution of 1 ? (c) Let 2 denote the time to the second failure. What is the distribution of 21 ? (d) As defined, N(t) is a pure birth process. Justify this claim. What is the relevant state space? For each state, what is the birth rate? Question 2. [12 marks]. A factory has produced m televisions which are for sale. Television i works for a random time Xi, where the Xi are independent and identically distributed with common distribution Exponential () where >0 is the rate parameter. Let N(t) denote the number of televisions that have failed by time t. (a) Calculate the probability P[N(t)=r], where r={0,1,,m}. (b) Let 1 denote the (random) time to the first failure. What is the distribution of 1 ? (c) Let 2 denote the time to the second failure. What is the distribution of 21 ? (d) As defined, N(t) is a pure birth process. Justify this claim. What is the relevant state space? For each state, what is the birth rate