= Solution
Continuity and strict increase make the <quantile function> $F^{-1}$ well-defined on $(0,1)$. The endpoints of the <uniform distribution> occur with <probability> zero and do not affect the construction. For every real $x$,
$$
\begin{aligned}
\Pr(X_i\leq x)
&=\Pr(F^{-1}(U_i)\leq x)\\
&=\Pr(U_i\leq F(x))=F(x).
\end{aligned}
$$
Thus each $X_i$ has <cumulative distribution function> $F$. Applying a fixed measurable function separately to <independent random variables> preserves their independence; explicitly, joint lower-tail probabilities factor as
$$
\Pr(X_1\leq x_1,\ldots,X_n\leq x_n)=\prod_{i=1}^nF(x_i).
$$
Consequently
$$
\boxed{X_1,\ldots,X_n\text{ are independent and identically distributed with CDF }F.}
$$
This is <inverse transform sampling>. Since $F^{-1}$ preserves order, it also sends the central <order statistic> to the central <order statistic>, justifying $M_X=F^{-1}(M_U)$.
Back to article page
