Given a random sample from a Geometric distribution with probability function $ P(X=x)=p*(1-p)^(x-1)$, $E(X)=1/p$, and $Var(X)=(1-p)/p^2$, I am trying to find the MLEs of $E(X)$ and $Var(X)$.
I have already found that the MLE of $p$ itself, $\hat{p}=1/ \bar{x}$, but I do not think finding the MLEs for the expectation and variance of X is as simple in this case as plugging p-hat into the formulas for expectation and variance.
Could you explain to me how this is done?
I think I've answered my own question.
By the Invariance Property of MLEs, if f(θ) is a one-to-one function of θ and if $\hat{θ}$ is the MLE for θ, then the MLE of f(θ) is given by $\hat{f(θ)}=f(\hat{θ})$.
So, it would not be inappropriate to plug $\hat{p}$ into $E(X)$ and $Var(X)$ to get the Maximum Likelihood Estimators.
This would give $\hat{E(X)}=\bar{X}$ and $\hat{Var(X)}=\bar{X^2}-\bar{X}$.
Is this correct?