. Assume that the lifetimes (measured from the beginning of use)of lightbulbs are i.i.d. random variables with distribution P(T ?k) = (k + 1)?? , k = 0, 1, 2, . . . , for some ? > 0. (Note thattime is measured in discrete units.) In a lightbulb socket in afactory, a bulb is used until it fails, and then it is replaced atthe next time unit. Let (Xn)n?0 be the irreducible Markov chainwhich records the age of the bulb currently in use in the socket(Xn = 0 at times when a bulb is replaced, corresponding to a newbulb). (a) Derive the transition probabilities of the chain. (b)For each value of ?, determine if the chain is positive recurrent,null recurrent, or transient.
