Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind spe...

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

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).