Please be as detailed as possible, and thank you so much this is urgent

1. A factory has produced N televisions for sale. Here N is a Poisson random variable with mean A. These televisions can fail to function at some time. Assume each tele- vision malfunctions independently. Let X.- denote the life time of the ith television produced by this factory, which is a positive random variable that follows an exponen- tial distribution with a rate parameter a. Let N (t) be the number of televisions which have failed to work by time t. [16 marks] (a) Find the mean of N (t) for each It > 0. [3 marks] (b) Identify the distribution of N (t) for each t > 0. [4 marks] (c) Does N (t) have independent increment? Explain your answer. Does your conclusion still hold if X7; follows a general distribution with a proper density function f, rather than an exponential distribution? [2 marks] ((1) Is (N (t),t 2 0) a nonhomogeneous Poisson process? Explain your answer. Does your conclusion still hold if X]- follows a general distribution with a proper density function f , rather than an exponential distribution? [3 marks] Let Y;- be a Gamma random variable with parameter (a, {3). Assume Y;- are independent. Dene a compound Poisson process N (t) m) = Z Y- (e) Find E(Y(t)) and cov(Y(s), Y(s + 15)) for each s, t > 0. [4 marks]