Unbiased estimator of $ 1 + \mu^{2}$ from a Normal population 
Question: If $ x_{1}, x_{2}, x_{3},...x_{n}$ is a random sample from a $Normal$ $population$ $N(\mu,1)$ then  what is the unbiased estimator of $ 1 + \mu^{2}$ ?

I began finding the mean and variance of $X$
$$E(X) = \mu \  , \ Var(X) = 1$$
And the expectation of $X^{2}$  is
$$  E(X^{2}) = 1 + \mu^{2} $$
Hence is the unbiased estimator of $ 1 + \mu^{2}$ =  $\frac{1}{n}\sum_{i=1}^{n} x_{i}^{2}$ ?
Is my method and solution correct?

Any function of the data is called an estimator. There is no such thing as "THE" estimator of a quantity. Various estimators can have different properties.
You have shown (correctly) that your estimator $\tfrac{1}{n}\sum x_i^2$ is unbiased for $1+\mu^2$. You could consider other estimators and they may have different properties (e.g. smaller or larger variance).
