Chapter 6.1, Problem 15E

Let X₁, X₂,..., X_n represent a random sample from a Rayleigh distribution with pdf

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

1. a. It can be shown that E(X²) = 2θ. Use this fact to construct an unbiased estimator of θ based on ${\displaystyle \sum X_i^2}$ (and use rules of expected value to show that it is unbiased).
2. b. Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions: