Fall 2011 Homework 5: Question 3
Let U be a Uniform r.v. on the interval (-1, 1) (be careful about minus signs).
(a) Compute $E(U)$, $Var(U)$, and $E(U^{4})$.
(b) Find the CDF and PDF of $U^{2}$. Is the distribution of $U^{2}$ 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). Consult iTunes course for detailed solutions.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."