# Find an estimator by using the method of moment

Let $X$ be a discrete random variable with density function: $$p(x;\theta)=\left(\frac{\theta}{2}\right)^{\lvert x\rvert}(1-\theta)^{1-\lvert x\rvert}$$ where $x\in\{-1,0,1\}$ and $\theta \in[0,1]$.

I have to find an estimator of $\theta$ by using the method of moment.

Now, the first and second moments are:

$\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=\theta$

Based on this, when a simple random sample of size $n$ is drawn:

$\hat \mu_1=\dfrac{1}{n}\sum_{i=1}^nx_i$ and $\hat \mu_2=\dfrac{1}{n}\sum_{i=1}^nx_i^2$

So, the estimator $\hat \theta$ is $$\hat \theta = \dfrac{1}{n}\sum_{i=1}^nx_i^2$$

Is that correct? :-)

Edit

I also have to prove that the estimator is unbiased and consistency. For first question: $$\mathbb{E}(\hat \theta)=\frac{1}{n}\mathbb{E}(X_1^2)+\frac{1}{n}\mathbb{E}(X_2^2)+\ldots+\frac{1}{n}\mathbb{E}(X_n^2)=\frac{1}{n}(\theta+\theta+\ldots+\theta)=\theta$$

thus the estimator is unbiased.

For consistency we need that $\lim_{n \to \infty}\mathbb{E}(\hat \theta)=\theta$ and $\lim_{n \to \infty}\mathbb{Var}(\hat \theta)=0$. Any help?