Finding the MLE's when $X_i \sim N(\mu,\sigma^2)$

Question

We have a random sample of size $n$ from a normal distribution, $X_i \sim N(\mu,\sigma^2)$. I have to find the MLE of the following:

1. $P[X > c]$ for arbitrary $c$
2. The $95^\text{th}$ percentile of $X$

Could someone give me tips how to handle these kind of questions? I have solved MLE questions before, so I understand the idea behind it, however I couldn't answer these two unfortunately.

So far, I have standardized the $P[X>c]$ and I know have $P[X>c] = 1 - F_X[\frac{c-\mu}{\sigma}]$. I guess I can now conclude something, but I do not know what.

Answer 1

Once you know that the MLE for the population mean is the sample mean $\overline x = (x_1+\cdots+x_n)/n$ and the MLE for the population variance is the (biased) sample variance $s^2 = \big( (x_1-\overline x)^2 +\cdots + (x_n-\overline x)^2\big)/n$ then the rest is a matter of applying the theorem on equivariance of MLEs.

(You often see this incorrectly called "invariance" rather than "equivariance.")

The value of $c$ for which $\Pr(X>c)=0.95$ is $\mu + \sigma\Phi^{-1}(0.95),$ where $\Phi$ is the standard normal c.d.f., so then equivariance tells you that $\widehat{c\,}_\text{m.l.e.} =\overline x + s\Phi^{-1}(0.95).$