A dam is used for generating electricity, controlling floods, and irrigating land. The dam has a capacity of 4 units of water. Assume that the volume of water stored in the dam is always an integer. Let X_n be the volume of water in the dam at the end of week n. During week n, a volume of water denoted by W_n flows into the dam. The probability distribution of W_n, which is an independent, identically distributed, and integer random variable, is given below:

At the end of each week, if the dam contains one or more units of water, exactly one unit of water is released. Whenever the inflow of water to the dam exceeds its capacity of 4 units, the surplus water is released over the spillway and lost. Construct the transition probability matrix.

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Volume W of water flowing into dam in week n, W = k 0 Probability, w = P(W = k) 1 2 3 4 5 or more W5 Wo W W2 W3 W4