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Suppose that $Y_{1}, \ldots, Y_{n}$ is a random sample from a $N\left(\theta, \sigma^{2}\right)$
population where both $\theta$ and $\sigma^{2}$ are parameters. Determine the method of
monent estimators $\hat{\theta}_{\text {MOM }}$ and $\hat{\sigma}_{\text {MOM}}^{2}$

$E(Y)=\theta$

$\sigma^{2}=E\left(Y^{2}\right)-E(Y)$

$E\left(Y^{2}\right)=\sigma^{2}+E(Y)$

$\hat{\mu}_{1}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}=E(Y)=\theta$

$\hat{\mu}_{2}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}^{2}=E\left(Y^{2}\right)=\sigma^{2}+ E(Y)$

$\theta=\frac{1}{n} \sum_{i=1}^{n} Y_{i} \quad (1)$

$\frac{1}{n} \sum_{i=1}^{n} Y_{i}^{2}=\sigma^{2}+\theta \quad (2)$

$\hat{\theta}_{MOM}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}$

$\hat{\sigma}_{MOM}^{2}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}^{2}-\frac{1}{n^{2}}\left(\sum_{i=1}^{n} Y_{i}\right)^{2}$

$\hat{\sigma}_{MOM}^{2}=\frac{n-1}{n} S^{2}$