(Solution) - Let F 0 1 be non decreasing right -(2025 Original AI-Free Solution)

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Let F : ( ? [0, 1] be non-decreasing, right continuous, F(- () = 0 and F(() = 1 (i.e., a d.f. of a r.v.), and let F-1 be defined by
F-1 (y) = inf {x ( (; F(x) ( y}, y ( [0, 1].
Next, consider the probability space ((, A, P) = ([0, 1], B[0, 1], P), where P = ( is the Lebesgue measure, and on (, define the function X by X (() = F-1((). Then show that X is a r.v. and that its d.f. is F.