4. 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.

Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution, with pdf f(x; )


x

2
e x2

/(2 2

) x 0 0 otherwise as a model for the X distribution.

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

b. Suppose
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
100)?

d. Give an expression for P(X  x).