[R] Two envelopes problem

Duncan Murdoch murdoch at stats.uwo.ca
Tue Aug 26 14:15:09 CEST 2008

On 26/08/2008 7:54 AM, Jim Lemon wrote:
> Hi again,
> Oops, I meant the expected value of the swap is:
> 5*0.5 + 20*0.5 = 12.5
> Too late, must get to bed.

But that is still wrong.  You want a conditional expectation, 
conditional on the observed value (10 in this case).  The answer depends 
on the distribution of the amount X, where the envelopes contain X and 
2X.  For example, if you knew that X was at most 5, you would know you 
had just observed 2X, and switching would be  a bad idea.

The paradox arises because people want to put a nonsensical Unif(0, 
infinity) distribution on X.  The Wikipedia article points out that it 
can also arise in cases where the distribution on X has infinite mean: 
a mathematically valid but still nonsensical possibility.

Duncan Murdoch

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