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Doug Rivers, professor of Political Sciences in Stanford, kindly sent me this email yesterday night:
The 2nd displayed equation in section 2.1.2 on p. 44 is garbled (it might be interpreted as saying that U and X have the same distribution). I think you intended:
And indeed we should have stated the implicit convention that u=F(x) before this equation, i.e.
(although U and X cannot be seen to have the same distribution as they live in different spaces).