Thank you!

3. Rayleigh Facts and Applications Let R have the Rayleigh density given by f (r) = re''2 for r > 0. Refer to Section 18.1 and Section 18.4 for ways in which this density arises. The point of Parts a, b, and d is for you not to reinvent wheels, and also for you to notice that math calculation can be reduced by using probability facts. The probabilistic approach to math is becoming increasingly important even in research. So follow the approach in the instructions; we will not give credit for other approaches. a) Let Z be standard normal. Use the expectation and variance of Z to find E(R) without any further integration. b) Use properties of R2 to find E(R2) without any further integration, and hence find Var(R). c) Let Z be standard normal. Find the density of |Z |. Don't try and use the formula. Just think about P(|Z | E dy) for positive y. d) Use facts about the Rayleigh density to find E(|Z |) without any further integration