Chapter2: Mathematics For Microeconomics
Section: Chapter Questions
Problem 2.6P
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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)
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2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82214613-ec36-4dd8-8d54-892cd4bff3e3%2F5ddec840-c7c3-45ff-a1dc-d2b91d11afc4%2Fhns2koj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:E[(X - m)+] ≤
√σ² + (m − µ)² - (m - µ)
-
2
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