Suppose that $X_{1}, \ldots, X_{n}$ are i.i.d. $N\left(\mu, \sigma^{2}\right)$ random variables.

(a) Compute the MLEs $\widehat{\mu}, \widehat{\sigma}^{2}$ for the unknown parameters $\mu, \sigma^{2}$.

(b) Give the definition of an unbiased estimator. Determine whether $\widehat{\mu}, \widehat{\sigma}^{2}$ are unbiased estimators for $\mu, \sigma^{2}$.