Let U be a Uniform r.v. on the interval (-1, 1) (be careful about minus signs).
(a) Compute , , and .
(b) Find the CDF and PDF of . Is the distribution of Uniform on (0, 1)?
Solution: Let G(t) be the CDF of U^2. G(t) = square root of t. G'(t) = 0.5t^-0.5 is the PDF of U^2. The distribution of U^2 is not Uniform on (0, 1) as the PDF is not a constant on this interval (it is an example of a Beta distribution, which is another important distribution in statistics and will be discussed later).
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"Mathematics is the logic of certainty, butstatistics is the logic of uncertainty."