Can I find MLE of probability of X greater than x such as find  MLE of $P(X>12)$ when $X_i$ ~ $N(\mu,\sigma^2) $
 A: As I'm sure you know (or can easily derive), the maximum likelihood estimator of $(\mu,\sigma^2)$ is
$$(\hat\mu,\hat\sigma^2) = \left(\bar X, \operatorname{Var}(X)\right)$$
where, as usual, $n\bar X = X_1+X_2+\cdots + X_n$ and, a little unusually,
$$\operatorname{Var}(X) = \frac{1}{n}\left((X_1-\bar X)^2 + (X_2-\bar X)^2 + \cdots + (X_n-\bar X)^2\right)$$
(notice the $n$ rather than the $n-1$ in the denominator).
A fundamental property of MLEs is that a maximum likelihood estimator of any function $f$ of the parameters equals $f$ applied to the MLEs of the parameters.  In this case, letting $\Phi$ be the standard Normal CDF
$$f(\mu,\sigma^2) = \Pr(X \gt 12) = 1 - \Phi\left(\frac{12-\mu}{\sqrt{\sigma^2}}\right).$$

Therefore the MLE of $\Pr(X \gt 12)$ is $$\hat f = 1 - \Phi\left(\frac{12-\hat\mu}{\sqrt{\hat\sigma^2}}\right).$$

Because maximum likelihood estimation is a procedure intended for relatively large sample sizes, here is a simulation based on 100,000 samples of size $n=240.$  The parameter was set to $(\mu,\sigma)=(5,3).$

You can see how the estimates are spread roughly evenly around the true value (marked as a red vertical segment), indicating this solution is a reasonable estimator.  Further simulations with ever larger sample sizes (I went up to $n=24000$ but limited the simulation to just a thousand iterations to keep the computation time down to a second or two) indicate this solution converges around the true value, as it ought.
#
# Specify the problem.
#
mu <- 5
sigma <- 3
threshold <- 12
n <- 240
#
# Draw samples.
#
set.seed(17)
n.sim <- 1e5
x <- matrix(rnorm(n.sim*n, mu, sigma), n.sim)
#
# Compute the MLEs.
#
mu.hat <- rowMeans(x)
sigma2.hat <- rowMeans((x - mu.hat)^2)
p.hat <- pnorm((threshold - mu.hat) / sqrt(sigma2.hat), lower.tail = FALSE)
#
# Plot the MLEs.
#
hist(p.hat, freq=FALSE, col="#f0f0f0", breaks=50,
     xlab=expression(hat(f)), cex.lab=1.25,
     main=expression(paste("Histogram of ", Pr(X>12))))
abline(v = pnorm(threshold, mu, sigma, lower.tail = FALSE), lwd=2, col="#d01010")

