An automatic machine produces (thousands of) units of a product per day. As increases, the

An automatic machine produces α (thousands of) units of a product per day. As α increases, the proportion of defectives, p, goes up according to the following probability density function

Each defective item incurs a loss of $50. A good item yields $5 profit.

(a) Develop a decision tree for this problem.

(b) Determine the value of α that maximizes the expected profit.

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f(p) = { apa-1, 0≤ p ≤ 1 otherwise 0,