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Unformatted text preview: Spring 2010 Pstat160b Example Random demand problem Suppose the demand X for some product is random with exponential distribution with parameter λ (say some fruits in tons). The manager of a store orders K units (can be continuous) for $ 2 per units (could actually be $2000 per tons), sell it back for $ 3. Demand that is not met is lost, and leftover product is lost. What is the expected profit? In this problem, we will need some properties of the exponential distribution. If X ∼ E ( λ ) then: f X ( x ) = λe- λx F X ( x ) = Z x λe- λu du = 1- e- λx P ( X > x ) = 1- F X ( x ) = e- λx Let Y be the profit. Then we have, computing profit=sales-expenses Y = 3 X- 2 K X ≤ K (all the demand is met) 3 K- 2 K = K X ≥ K (all the supply is sold) Therefore, the profit is a function of the demand, Y = g ( X ) where the function g is defined by g ( x ) = 3 x- 2 K x ≤ K K x ≥ K To compute the expected profit Ep , we use the formula Ep = E [ g ( X )] = Z ∞-∞...
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- Spring '10
- Probability theory, Exponential distribution, dx, xλe−λx dx