The sample r.v. $X_1, X_2,\ldots, X_n$ i.i.d. $N(\mu, \sigma^2)$, find the moment estimator of $P\{X>1\}$, where $X\sim N(\mu, \sigma^2)$.
Here is my answer. I want to know whether it is right, thx.
Note that $\hat \mu = \bar X, \hat{\sigma^2} = S^2.$
$$
\begin{aligned}
P\{X>1\} &= \int_1^\infty \frac1{\sqrt{2\pi}\sigma} {\rm e}^{\frac{(x-\mu)^2}{2\sigma^2}} {\rm d}x \\
&= 1 - \Phi(\frac{1-\mu}{\sigma}) \\
\end{aligned}
$$
where $\Phi(x) = \int_{-\infty}^x \frac1{\sqrt{2\pi}} {\rm e}^{-\frac{t^2}{2}} {\rm d}t$.
Then I substitute $\hat \mu$ and $\hat\sigma$ to right hand side, hence moment estimator is
$$
\hat {P\{X>1\}} = 1 - \Phi(\frac{1 - \bar X}{S})
$$
Thanks to the comment.
My textbook gives $\hat{\sigma^2} = \frac{1}{n-1}\sum_{i=1}^n(X_i - \bar X)^2$.