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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](http://l.wordpress.com/latex.php?latex=P(U+%5Cleq+u)+=+P%5BF(X)+%5Cleq+u%5D+=+P%5BF%5E%7B-1%7D(F(X))+%5Cleq+F%5E%7B-1%7D(u)%5D+=+P%5BX+%5Cleq+F%5E%7B-1%7D(u)%5D+=+F%5BF%5E%7B-1%7D(u)%5D+=+u&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "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")
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)](http://l.wordpress.com/latex.php?latex=P(U+%5Cleq+F(x))=P%5BF(X)+%5Cleq+F(x)%5D=P%5BX+%5Cleq+x%5D=F(x)&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "P(U leq F(x))=P[F(X) leq F(x)]=P[X leq x]=F(x)")
(although U and X cannot be seen to have the same distribution as they live in different spaces).
Filed under: Books, R, Statistics Tagged: Introducing Monte Carlo Methods with R, typos
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