[miloshh]

"use of an impossible probability distribution" - this sounds interesting. Which exact distribution do you mean? To me it seems (though I might be wrong) that the distributions are perfectly valid.

You have a pair of random variables (X, Y) that take values (100, 200) or (200, 100) with equal probability. Then E[X] = E[Y] = 150, there is no question about that. Also, E[X/Y] = 1.25, there is also no question. The only question is why E[X] is useful and E[X/Y] is not useful for our decision making - and I honestly don't know why.

[brazzy]

If it were (100, 200) or (200, 100) with equal probability, you'd switch iff the first envelope contains 100. Nothing interesting about that scenario.

The problem states that (and is only interesting because) one of the envelopes contains twice as much as the other, but not how much, so that the amount in the first envelope tells you nothing.

This is actually impossible because there is no information about how the amount (let's say the bigger of the two) is distributed, which usually implies uniform distribution. But a uniform distribution is only possible if you assume an upper limit (otherwise, what's the expected value?). If there is an upper limit, the question again becomes quite easy (you switch if the first envelope contains less than half the upper limit).

[miloshh]

"Nothing interesting about that scenario." OK, but let's say you're not allowed to see the contents of the envelope you picked.

In this case, the argument that by switching you get 125% on average still holds!

I'd claim that this is even more interesting and paradoxical than the scenario you're talking about, where your explanation is correct (the distribution of the bigger value is crucial for your decision, and you know nothing about it).