Question: Consider a specific kind of part needed for the operation of a certain machine (e.g., the water pump of a car). When the part breaks

Consider a specific kind of part needed for the operation of a certain machine (e.g., the water pump of a car). When the part breaks down, it is replaced by a new one.

The probability that a new part will last for exactly n days is rn,n = 1, 2, ... . Let the state of the system be defined as the age of the part currently in the machine. Find the transition probability matrix of the resulting chain.

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