I learned a new variation of the two envelopes problem and it turns out I hate math and also infinity is banned.
Am I missing something? The distribution for X looks like just a distraction added to the problem. Initial selection is still independent.
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It's really not, because once you've looked in your envelope you can compare it to the initial distribution of X. If the distribution of X was, say, $1, $2, and $4 with equal probability, you'd know what to do if you saw $8.
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The issue is that if you decide you should *always* switch regardless of what you see, then looking in the envelope didn't matter and thus it shouldn't matter which envelope you get.
End of conversation
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