Consider a simple sample $X_{1},X_{2},\ldots,X_{n}$ whose distribution is given by $X\sim Exp(\lambda)$.
(a) Determine an estimator for $\lambda$ according to the method of moments.
(b) Determine another estimator for $\lambda$ different from the previous one.
(c) Determine an estimate of $\textbf{P}(X\geq 1)$ in accordance to the method of moments.
MY ATTEMPT
As to the first case, we have \begin{align*} \frac{1}{\lambda} = \textbf{E}(X) \Rightarrow \hat{\lambda} = \frac{n}{\displaystyle\sum_{k=1}^{n}X_{k}} \end{align*}
As to the second case, we have \begin{align*} \frac{1}{\lambda^{2}} = \textbf{Var}(X) = \textbf{E}(X^{2}) - \textbf{E}(X)^{2} \Rightarrow \hat{\lambda} = \left[\frac{1}{n}\sum_{k=1}^{n}X^{2}_{k} - \left(\frac{1}{n}\sum_{k=1}^{n}X_{k}\right)^{2}\right]^{-1/2} \end{align*}
EDIT
In the third case, we have \begin{align*} \textbf{P}(X\geq 1) = 1 - \textbf{P}(X < 1) = 1 - \exp(-1\times\lambda) = 1 - \exp(-\lambda) \end{align*}
Now it suffices to substitute $\lambda$ for any of the above expressions.
Could someone double-check my reasoning? Thanks in advance!