1. Decision Theory Consider the following random variable X. For positive integers k, Pr[X = 2k] = 1/2k. One interpre- tation of this random variable is as the following lottery. Toss a fair coin until it comes up heads. If it comes up heads on the kth toss, the lottery payout is 2k dollars. (a) Prove that E[X] = ∞. (b) It seems unreasonable to believe that someone would pay an arbitrarily large amount of money to participate in this lottery. We discussed one potential reason for this: that people are risk-averse and have a concave utility for money. Suppose that my utility for r dollars from the lottery is log₂ r. What is my expected utility for this lottery? (Hint: You should be able to express the value of the lottery as an infinite sum. You may then consult references for the value of that sum.)
1. Decision Theory Consider the following random variable X. For positive integers k, Pr[X = 2k] = 1/2k. One interpre- tation of this random variable is as the following lottery. Toss a fair coin until it comes up heads. If it comes up heads on the kth toss, the lottery payout is 2k dollars. (a) Prove that E[X] = ∞. (b) It seems unreasonable to believe that someone would pay an arbitrarily large amount of money to participate in this lottery. We discussed one potential reason for this: that people are risk-averse and have a concave utility for money. Suppose that my utility for r dollars from the lottery is log₂ r. What is my expected utility for this lottery? (Hint: You should be able to express the value of the lottery as an infinite sum. You may then consult references for the value of that sum.)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 54E
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![1. Decision Theory
Consider the following random variable X. For positive integers k, Pr[X = 2k] = 1/2k. One interpre-
tation of this random variable is as the following lottery. Toss a fair coin until it comes up heads. If it
comes up heads on the kth toss, the lottery payout is 2k dollars.
(a) Prove that E[X] = ∞o.
(b) It seems unreasonable to believe that someone would pay an arbitrarily large amount of money to
participate in this lottery. We discussed one potential reason for this: that people are risk-averse
and have a concave utility for money. Suppose that my utility for x dollars from the lottery is
log₂ r. What is my expected utility for this lottery? (Hint: You should be able to express the
value of the lottery as an infinite sum. You may then consult references for the value of that sum.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F672bf286-8abe-4b07-9ca1-0d5b2612956c%2F28dbd6c1-9144-4177-957c-7e3073850925%2Fkzc20va_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Decision Theory
Consider the following random variable X. For positive integers k, Pr[X = 2k] = 1/2k. One interpre-
tation of this random variable is as the following lottery. Toss a fair coin until it comes up heads. If it
comes up heads on the kth toss, the lottery payout is 2k dollars.
(a) Prove that E[X] = ∞o.
(b) It seems unreasonable to believe that someone would pay an arbitrarily large amount of money to
participate in this lottery. We discussed one potential reason for this: that people are risk-averse
and have a concave utility for money. Suppose that my utility for x dollars from the lottery is
log₂ r. What is my expected utility for this lottery? (Hint: You should be able to express the
value of the lottery as an infinite sum. You may then consult references for the value of that sum.)
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