A machine is made up of two components that operate independently. The lifetime T i (in days) of component i has an exponential distribution with parameter λ i , for i = 1,2. Suppose that the two...

A machine is made up of two components that operate independently. The lifetime
T_i
(in days) of component
i
has an exponential distribution with parameter λ_i, for i = 1,2.

Suppose that the two components are placed in parallel and that λ₁
= λ₂
= ln2. When the machine breaks down, the two components are replaced by new ones at the beginning of the following day. Let
Xn
be the number of components that operate at the end of
n
days. Then the stochastic process {Xn,n = 0 , 1 , . . .} is a Markov chain. Calculate its one-step transition probability matrix.