Let $X_1,\ldots,X_n$ ~ Poi($\theta$). Calculate the estimatorios of moments for $\theta$ in the next cases: Remember that $E(X_1)=Var(X_1)=\theta$
(a) $\hat{\theta}_1$ by equating the first non-central theoretical moment with the first non-central empirical moment.
(b) $\hat{\theta}_2$ by equating the theoretical variance with the empirical variance.
(c) $\hat{\theta}_3$ by equating the second non-central theoretical moment with the second non-central empirical moment.
I just want to know if i did it correctly. It seems too easy.
(a) $\hat{\theta}_1=\overline{x}=\frac{1}{n}\sum^{n}_{i=1}x_i$
(b) $\hat{\theta}_2=\overline{x}=\frac{1}{n}\sum^{n}_{i=1}(x_i-\overline{x}_1)^2$
(c) $\hat{\theta}_3^2+\hat{\theta}_3=\frac{1}{n}\sum^{n}_{i=1}x_i^2$
As for the empirical variance, $\mu_1$ cannot be part of the formula, since actually $\mu_1=\theta$, and then it is not an estimator for $\theta$. You should use $$\hat \theta_2=\frac1n \sum(x_i-\bar x)^2,$$ instead.
As for $\hat \theta_3$, that's just how you start; now solve for $\hat \theta_3$ (it is a quadratic equation), and then you'll have the estimator.