Question: Show that if a r.v. X has a continuous d.f. F(x), then the distribution of the r.v. Y = F(X) coincides with the distribution uniform

Show that if a r.v. X has a continuous d.f. F(x), then the distribution of the r.v. Y = F(X) coincides with the distribution uniform on [0,1]. (The r.v. Y may not assume values 0 and 1, but for the uniform distribution, the probabilities of these values equal zero anyway.) Show that the continuity condition is necessary.

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