Chapter 4.1, Problem 4E

Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (J. of Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution, with pdf

     $f(x;\theta )=\{\begin{array}{ll}\frac{x}{{\theta }^2}\cdot e^{-x^2/(2{\theta }^2)}\hfill & x>0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$

as a model for the X distribution.

a. Verify that f (x; θ) is a legitimate pdf.

b. Suppose θ = 100 (a value suggested by a graph in the article). What is the probability that X is at most 200? Less than 200? At least 200?

c. What is the probability that X is between 100 and 200 (again assuming θ = 100)?

d. Give an expression for P(X ≤ x).