ProbabilityMethods Of Point Estimation Problem

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 $$\hat{\sigma}_{\text {MOM}}^{2}$$ 

$$E(Y)=\theta$$

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

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

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

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

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

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

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

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

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