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

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)

(although U and X cannot be seen to have the same distribution as they live in different spaces).

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