We have an infinite supply of light bulbs, and Zi is the lifetime of the i-th light bulb. {Zi, i ≥...

Question

We have an infinite supply of light bulbs, and Zi is the lifetime of the i-th light

bulb. {Zi, i ≥ 1} is a sequence of iid discrete random variables with common pmf

P(Zi = k) = pk, k = 1, 2, 3, ,

with ∞

k=1 pk = 1. At time zero, the first light bulb is turned on. It fails at time Z1,

when it is replaced by the second light bulb, which fails at time Z1 + Z2, and so on.

Let Xn be the age of the light bulb that is on at time n. Note that Xn = 0, if a new

light bulb was installed at time n. Show that {Xn, n ≥ 0} is a DTMC and compute

its transition probability matrix.

Aditi · Accepted Answer

(
1
)
P[Xn+1 = 0  Xn = j] = P[Zi(n) = j + 1	Zi(n) > j] =	.Σ|	|		∞
k=j+1 pk

We have an infinite supply of light bulbs, and Zi is the lifetime of the i-th light bulb. {Zi, i ≥ 1} is a sequence of iid discrete random variables with common pmf P(Zi = k) = pk, k = 1, 2, 3, , with...