I have:
$$ X_{1}, ...,X_{n} \sim Exp(\beta) $$
with the probability density function: $f(x, \beta) = \frac{1}{\beta} \exp(-\frac{x}{\beta})$ where $x > 0$.
I consider: $\theta = P(X_{1} > 1)$. My question is to find the maximum likelihood estimator of $\theta$(denoted as $\hat \theta_{MLE}$).
This is my attempt:
Since I know that $P(X > 1) = \exp(-\frac{1}{\beta})$ (survival time function), I get the likelihood function that:
$$ L(\beta) = \exp(-\frac{n}{\beta}) $$
and log-likelihood function:
$l(\beta) = -\frac{n}{\beta}$
and $l'(\beta) = \frac{n}{\beta^2}$
Which fails the calculus procedure to find MLE of $\beta$.
It seems to me that: Is $\beta = max\{ X_1, ..., X_n \} $?