3. Assume that the lifetimes (measured from the beginning of use) of lightbulbs are i.i.d random...

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3. Assume that the lifetimes (measured from the beginning of use) of lightbulbs are i.i.d random variables with distribution P(T k)(k1)-, ., k 0,1, 2, for some B 0. (Note that time is measured in discrete units.) In a lightbulb socket in factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (XT)0 be the irreducible Markov chain which records the age of the bulb currently in use in the socket (Xp a 0 at times when a bulb is replaced, corresponding to a new bulb) (a) Derive the transition probabilities of the chain (b) For each value of B, determine if the chain is positive recurrent, null recurrent, or transient 3. Assume that the lifetimes (measured from the beginning of use) of lightbulbs are i.i.d random variables with distribution P(T k)(k1)-, ., k 0,1, 2, for some B 0. (Note that time is measured in discrete units.) In a lightbulb socket in factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (XT)0 be the irreducible Markov chain which records the age of the bulb currently in use in the socket (Xp a 0 at times when a bulb is replaced, corresponding to a new bulb) (a) Derive the transition probabilities of the chain (b) For each value of B, determine if the chain is positive recurrent, null recurrent, or transient