Introducing Monte Carlo Methods with R [precision]
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:
![P(U \leq u) = P[F(X) \leq u] = P[F^{-1}(F(X)) \leq F^{-1}(u)] = P[X \leq F^{-1}(u)] = F[F^{-1}(u)] = u](https://s0.wp.com/latex.php?latex=P%28U+%5Cleq+u%29+%3D+P%5BF%28X%29+%5Cleq+u%5D+%3D+P%5BF%5E%7B-1%7D%28F%28X%29%29+%5Cleq+F%5E%7B-1%7D%28u%29%5D+%3D+P%5BX+%5Cleq+F%5E%7B-1%7D%28u%29%5D+%3D+F%5BF%5E%7B-1%7D%28u%29%5D+%3D+u&bg=000000&fg=B0B0B0&s=0&c=20201002)
And indeed we should have stated the implicit convention that u=F(x) before this equation, i.e.
![P(U \leq F(x))=P[F(X) \leq F(x)]=P[X \leq x]=F(x)](https://s0.wp.com/latex.php?latex=P%28U+%5Cleq+F%28x%29%29%3DP%5BF%28X%29+%5Cleq+F%28x%29%5D%3DP%5BX+%5Cleq+x%5D%3DF%28x%29&bg=000000&fg=B0B0B0&s=0&c=20201002)
(although U and X cannot be seen to have the same distribution as they live in different spaces).
Xi'an's Og 
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