If $U \sim Uni([0,1])$, does $U| U\ge 0.5 \sim Uni([0.5,1])?$ If it's false, why? and if it's true, can it be generalized for any Uniform Distribution?
2026-03-29 23:40:46.1774827646
Uniform distribution and conditional distribution
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Yes. You can prove it mathematically by calculating the PDF, but it is more enlightening to think about it intuitively:
A uniform distribution means that all points in the interval are as likely. When conditioning on the event of being in a particular domain, you do not change the ratio between the points, just scale all of them uniformly. Hence, the distribution is still uniform in the new domain.