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stilyo
Topic Author
Posts: 169
Joined: January 12th, 2009, 6:31 pm

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

Hi all -

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

Can we find an upper bound for 𝐸[𝑚𝑖𝑛(𝑋,𝑌)] that is better than 𝑚𝑖𝑛(𝐸[𝑋],𝐸[𝑌]) with the information given without knowing the distributions? Thanks for your help!

Alan
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Joined: December 19th, 2001, 4:01 am
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Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

One thought: maybe from here you can figure out how to sample from these distributions, so as to perhaps get a better estimate.

bearish
Posts: 6419
Joined: February 3rd, 2011, 2:19 pm

Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

I think the answer is no. Consider the counterexample (to any clever “proof” you may think of), where X equals 1 or 2 with equal probabilities and Y equals 3 or 4 with equal probabilities.

stilyo
Topic Author
Posts: 169
Joined: January 12th, 2009, 6:31 pm

Re: Upper Bound - Expectation of a Minimum of 2 Independent Random Variables

Thank you! Both your responses make sense - you can’t improve the inequality in general but certainly there are ways to optimize this for specific distributions given the information we have.