Question: Let $X_{1}, X_{2}, \dots, X_{n}$ be distributed with probability density function $f(x;\theta)$. Find the maximum likelihood estimator for $\theta$, if $$f(x;\theta) := \begin{cases} e^{-(x - \theta)} & \text{if } x \geq \theta \\ 0 & \text{otherwise} \end{cases}$$
Is this a case of computing the maximum likelihood function $\mathcal{L}(\theta)$ as the product of $f(x;\theta)$ and $\mathbb{P}(x \geq \theta)$ using the cumulative distribution function? For example like this?
$$ \mathcal{L}(\theta) = \prod_{i=0}^{n} f(x_{i};\theta)\mathbb{P}(x_{i} \geq \theta)$$
If not, how do we deal with the dependency on the value of $x$ in the definition of $f$?
By definition, you simply need to multiply $f(X_1;\theta)$, $f(X_2;\theta)$, $\ldots$, $f(X_n;\theta)$: $$\mathcal{L}(\theta) = \prod_{i=0}^{n} f(X_{i};\theta)=\begin{cases} e^{-(\sum_{i=1}^nX_i - n\theta)} & \text{if } X_1 \geq \theta,\ldots,X_n\geq \theta \\ 0 & \text{otherwise} \end{cases}=\begin{cases} e^{n\theta-\sum_{i=1}^nX_i} & \text{if } \theta\leq \min\{X_1,\ldots,X_n\}\\ 0 & \text{otherwise} \end{cases}$$ You can сonstruct a graph of this function with respect to $\theta$ and find MLE.