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I have a sample of data points independently sampled from a uniform distribution with a density function $f(x)=\frac{1}{a}, 0\leq x \leq a$.

I need to use the method of moments to estimate $a_{mom}$.

My idea is to just calculate the variance, then multiply by 12 and take the square root so I get $b-a$. Then I'm kind of stumped what to do further.

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An easier method-of-moments estimator arises from considering the mean. Notice that $E(X) = \frac{a}{2}$, so that one can take $$\hat{a}_{MoM} = 2 \overline{X}$$ as an estimator. Your approach would also work, since we have $$\operatorname{Var}(X) = \frac{a}{12}$$ so that $$a_{MoM}^\prime = 12 \hat{S}^2$$ where $\hat{S}^2$ is the sample variance.

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