Problem statement: Based on a rample sample of size $n$ from a normal distribution, $X \text{~} N(\mu, \sigma^2)$, find the MLEs of the following:
(a) $P[X>c]$ for arbitrary $c$,
(b) The 95th percentile of $X$.
My question: I just don't understand how to estimate quantities like these. I'm used to estimating unknown parameters that are part of density functions (e.g. $\theta$ in $f(x; \theta))$. Could you help me to get started here?
(a) $\mathbb{P} \left( x > c \right) = \mathbb{P} \left( \frac{x - \mu}{\sigma} > \frac{c - \mu}{\sigma} \right) = 1 - F_{\xi} \left( \frac{c - \mu}{\sigma} \right), \ $ where $\xi \sim \mathcal{N}(0, 1). \ $ $F_{\xi}(x) = \frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{x} e^{-\frac{t^2}{2}} \ dt $.
So the estimate $\hat{\mathbb{P}(x)} = 1 - F_{\xi} \left( \frac{c - \hat \mu}{\hat \sigma} \right), \ $where $\hat \mu = \bar x, \ \hat \sigma = \frac{1}{n}\sum\limits_{i=1}^n (x_{i} - \bar x)^2$
(b) Just put c = 0.05 in the estimate in (a).