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$$\large fx(x;a)=\frac{a}{x^{a+1}} , x \gt 1$$

where $a \gt1$ is the parameter of the distribution.

Find the estimator for $a$ using the method of moments.

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closed as off-topic by Siong Thye Goh, StubbornAtom, Ernie060, Lee David Chung Lin, Shailesh May 11 at 3:08

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The mean is $\Bbb E X=\int_1^\infty ax^{-a}dx=\frac{a}{a-1}$. Given an empirical estimator $\hat{\mu}$ of this mean, MOM's estimator $\hat{a}$ of $a$ satisfies $$\hat{\mu}=\frac{\hat{a}}{\hat{a}-1}\implies\hat{a}=\frac{\hat{\mu}}{\hat{\mu}-1}.$$

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