Research Seminars @ Illinois

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Tailored for undergraduate researchers, this calendar is a curated list of research seminars at the University of Illinois. Explore the diverse world of research and expand your knowledge through engaging sessions designed to inspire and enlighten.

To have your events added or removed from this calendar, please contact OUR at ugresearch@illinois.edu

Statistics Seminar - Stephen Portnoy (Illinois) "The 2-envelope paradox"

Event Type: Seminar/Symposium
Sponsor: Department of Statistics
Location: 2233 Everitt Lab
Date: Jun 13, 2024 2:00 pm
Views: 53
Originating Calendar: Department of Statistics Event Calendar; Title: The 2-envelope paradox

The 2-envelope paraox is a classical probability conundrum. Two players each receive an envelope: one containing the amount a and the other 2a. By turns, each player may either keep the amount received or switch envelopes. The conundrum is that if a player assumes the envelopes are equally likely, it is always best to switch, which seems paradoxical. However, having observed one value, the problem becomes essentially one of hypothesis testing based on a single observation, and the conditional probabilities will generally fail to be equally likely. Thus, the player must condition on the observed value, making the problem one of standard statistical inference, and not paradoxical. Here we take a general nonparametric approach and consider finding the envelope containing the larger value of the expectation of any specified function, v(x). The basic result is that if v(x) is bounded, then there is a randomized rule (called a "Blackwell pointer" by some colleagues) under which the success probability for choosing the larger envelope is greater than 1/2, uniformly over the set of all distribution-pairs. Other criteria are considered and counter-examples are presented to show that more general conditions will not provide success probabilities greater than 1/2.

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