How to derive the Method of Moments estimator of mu, using the second moment of X, when X is norm distributed?

Question

If $X_1,\ldots,X_n$ follow a normal distribution, where the variance $\sigma$ is given, how can you derive the MME of the mean $\mu$ using the second moment?

Answer 1

Note that the second moment of $X$ is $$E[X^2] = \sigma^2 + \mu^2,$$ so by subtracting the known variance from the second moment, you can find the square of the mean.

To this end, compute an estimate of the second moment $$\hat s = \frac {1}{N} \sum_{i=1}^N X_i^2,$$ then find the estimate of the mean with $$\hat \mu = \sqrt{\hat s -\sigma^2}.$$