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.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![# Decision Theory
Consider the following random variable \( X \). For positive integers \( k \), \( \Pr[X = 2^k] = 1/2^k \). One interpretation 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 \( k \)-th toss, the lottery payout is \( 2^k \) dollars.
(a) **Prove that \( E[X] = \infty \).**
(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 \( \log_2 x \). 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:# Decision Theory
Consider the following random variable \( X \). For positive integers \( k \), \( \Pr[X = 2^k] = 1/2^k \). One interpretation 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 \( k \)-th toss, the lottery payout is \( 2^k \) dollars.
(a) **Prove that \( E[X] = \infty \).**
(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 \( \log_2 x \). 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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