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Prove the following result in the derivation of the asymptotic optimality of the bid-price control policy. Let X be a random variable with finite first and second order moments, i.e., E[X] = µ < +∞, E[(X − E[X])^2] = σ^2 < +∞. Let x+ = max{x, 0}. Prove the following: ∀m ∈ R,
E[(X − m)+] ≤ (√ (σ2 + (m − µ)^2) − (m − µ))/2.
Hint: You can without loss of generality let m = 0
![E[(X - m)+] ≤
√σ² + (m − µ)² - (m - µ)
-
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- I need help with #111 please20) This decision tree represents the expected profits and the standard deviations associated with three decisions facing a mobile phone producer (All figures are in millions of dollars). The root node (the one on the left) represents the decision of whether to produce the phones in China or North America. The second pair of nodes represent the decision of whether to market the phones in China or North America; and the final nodes represent the choice of selling price: if the phones are sold in China, they will be sold for either $30 or $40, whereas if they are sold in North America, they will be sold for either $40 or $50. Based on the calculated values, what is the company's best strategy? Mean $30 1.250, sd 750 Sell China $40 Mean 375, sd = 625 Make in China Mean= -1,000, sd = 1,500 $40 Sell NA S50 Mean =-1,500, sd = 1,250 Sell China $30 $40 Make in NA Mean 1,250, sd.- 2,137 Sell NA $40 S50 Mean --250, sd-1.639
