$\displaystyle X_1 ... X_n$ is random sample from Rayleigh distribution $\displaystyle f(x;\Theta)$

1. Show that $\displaystyle E(X^2) = 2\Theta$ and than construct unbiased estimator of parameter $\displaystyle \Theta$ based on $\displaystyle \sum_{k=1}^n X_k^2$

2. Estimate parameter $\displaystyle \Theta$ from following $\displaystyle n=10$ observations:

Code:

16.88 10.23 4.59 6.66 13.68 14.23 19.87 9.40 6.51 10.95

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**1.** I have just plugged in 2 theta in Rayleigh's variance formula and it evaluates to true, but I'm not sure about correct way of constructing unbiased estimator $\displaystyle \hat{\Theta}$

$\displaystyle Var(X)=E(X^2) - E^2(X)$

$\displaystyle Var(X)=2\Theta - E^2(X)$

$\displaystyle \frac{4-\pi}{2}\Theta=2\Theta - \sqrt{\Theta\cdot\frac{\pi}{2}}^2$

$\displaystyle 4\Theta - \pi\Theta = 2(2\Theta - \frac{\pi\Theta}{2})$

$\displaystyle \Theta = \Theta $

**2.** I need help with this one