I'm asked to define $\text{Var}(\lambda_{MLE})$, i.e. where $\lambda_\text{MLE}$ is an estimator.
Additionally I'm asked to find the MLE of variance.
What's the difference between these? How is each found?
Is MLE of variance the same as the "variance estimator"
$$S^2=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2 ?$$
If yes, then how has this been derived?
$Var(\lambda_{MLE})$ is the true variance associated with the sampling distribution of your estimator $\lambda_{MLE}$. This value is generally unknown as it is likely to depend on unknown parameters in your model. For example, take the simple case of $y_1,...,y_n \overset{iid}{\sim} N(\mu,\sigma^2)$. The MLE of $\mu$ is $\bar{y}$. We know that $Var(\bar{y}) = \frac{\sigma^2}{n}$, but since $\sigma$ is generally unknown, we can't actually calculate this value in theory.
To get an actual estimate of the variance, we can plug in a consistent estimator of the unknown parameters such as the MLE of $\sigma$. Therefore, the MLE of the variance is $\hat{Var}(\bar{y}) = \frac{\hat{\sigma}^2}{n}$.
In short, the main difference is the variance of an estimator is a theoretical quantity that is not generally available. The MLE of the variance, however, is a numerically calculated estimate of this theoretically unknown quantity that can be used for the purposes of statistical inference.
