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stilyo
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Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

April 23rd, 2020, 2:08 pm

Hi all -

I have two independent (but not necessarily identically distributed) random variables: ๐‘‹>0,๐‘Œ>0. All of their moments are known but we don't know the distributions.

Can we find an upper bound for ๐ธ[๐‘š๐‘–๐‘›(๐‘‹,๐‘Œ)] that is better than ๐‘š๐‘–๐‘›(๐ธ[๐‘‹],๐ธ[๐‘Œ]) with the information given without knowing the distributions? Thanks for your help!
 
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Alan
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Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

April 23rd, 2020, 8:23 pm

One thought: maybe fromย here you can figure out how to sample from these distributions, so as to perhaps get a better estimate.
 
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bearish
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Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

April 25th, 2020, 12:15 am

I think the answer is no. Consider the counterexample (to any clever โ€œproofโ€ you may think of), where X equals 1 or 2 with equal probabilities and Y equals 3 or 4 with equal probabilities.
 
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stilyo
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Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

May 14th, 2020, 1:07 am

Thank you! Both your responses make sense - you canโ€™t improve the inequality in general but certainly there are ways to optimize this for specific distributions given the information we have.