Answered: E[(X - m)+] ≤ √σ² + (m − µ)² - (m - µ)…

Business

Economics

E[(X - m)+] ≤ √σ² + (m − µ)² - (m - µ) - 2

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

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)
-
2

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