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 p. Let th} be the number of televisions which have failed to work by time t. [1'6 marks] {a} Find the mean of N\") for each 1 :5 . [3 marks] [b] Identify the distribution of N} for each t :5 . [4 marks] (c) Does th) have independent increment? Explain your answer. Does your conclusion still hold if X1- follows a general distribution with a proper density function I. rather than an exponential distribution? [2 marks] [d] Is {th},t 3 [I] 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 (o, ,3}. Assume Y; are independent. Dene a compound Poisson process Nit} th} = 2 i1. i=1 {e} Find E[Y{t}} and cov{Y[s),Y{s + t)} for each s?! 2:: I]. [4 marks]