In t his problem \Ve prove a generalization of T heor em
Chapter 6, Problem 6.38(choose chapter or problem)
In t his problem \Ve prove a generalization of T heor em 6.5. Given a random variable X wit h CDF Fx(x), define F(11) = min {x lFx(x) >11,}. This problem proves t hat for a continuo_us uniform (0, 1) random variable U, X = F(U) has CDF F x(x) = Fx(x) . (a) Sho\v t hat when Fx(x) is a continuous, strictly increasing function (i.e., X is not mixed , Fx(x) has no j ump discontinuities, and Fx(x) h as no "flat" intervals (a,b) 'vhere Fx(x) = c for - - 1 a, < x < b), t hen F( 'IL) = F x ( 'IJ,) for 0 < 'IJ, < 1. (b) Show t ha! if Fx(:i;) has a jump at 1; = xo, t hen F( 'IJ,) = :i;o for all 11, in t he interval( c) Prove t hat X = F(U) has CDFFx(x) = Fx(:i;).
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