There is an infinite supply of light bulbs, and Zi is the lifetime of the ith light bulb.

Show that {Xn, n ? 0} is a DTMC and display its transition probability matrix.

There is an infinite supply of light bulbs. and Z; is the lifetime of the ith light bulb. {EL-1i :5 1} is a sequence of i.i.d. discrete random variables with HZ} = k] = pk. I: = l, 2, . . .. where pg 33 CI and 23:13:\": =1. At time I}. the first light bulb is turned on. It fails at time Z1; then it is replaced by the second light bulb which fails at time 31 + Z; and so on. Now let X\" be the age of the light bulb that is on at time 11. Note that Xn = if a failure has taken place at time H. Show that {an E D} is a DTMC and display its transition probability matrix