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RobinZ comments on Calibration for continuous quantities - Less Wrong
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Comments (13)
You shouldn't need to do any integrals to show that the PIT gives a uniform distribution. Suppose Pr(X <= x) = p; then (assuming no jumps in the cdf) the PIT maps x to p. In other words, writing P for the random variable produced by the PIT, Pr(P <= p) = p, so P is uniform.
Pr(X <= x) is an integral, but I find this explanation clearer than the one in the OP. Upvoted.