Question

Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. Use 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 = 134. What is the probability that X is at most 200? Less than 200? At least 200? (Round your answer to four decimal places.)

 (c) What is the probability that X is between 100 and 200 (again assuming θ = 134)? (Round your answer to four decimal places.)

 (d) Give an expression for P(X ≤ x).

Answer 1

a) To show that is a legitimate pdf. We need to show that

So,  

let

Hence,  

b) Before finding the probabilities let us find the cumulative distribution function:

F(x)=

again let

if , then probability that X is at most 200

probability that X is less than 200

, since it is a continuous distribution the equality doesn't matter.

Probability that x is at least 200

c)

Probability that X is between 100 and 200.

d) Already obtained earlier.