Parameter estimation on exponential distribution from a bounded subset of that distribution I have a random variable that is exponentially distributed with some $\lambda$. I'm sampling observations from this variable, but I'm limited to observing only those that are less than some maximum value $m$ ($m$ is a known value). In many cases, the event is not observed during the interval $[0,m]$, meaning either it would have occured later or not at all. From this, I want to do parameter estimation on the overall distribution.
I believe the PDF of this distribution should be $\frac{\lambda e^{-\lambda x}}{1-e^{-\lambda m}} (0\le x \le m)$ (the denominator just being the CDF of the exponential from 0 to $m$), and from there I can do MLE to get $\lambda$.
My questions are twofold: (1) is this a valid estimation to perform, and (2) is there existing terminology to talk about this sort of sampling (from a subset of the support of the original distribution)? This seems too simple to not have a name already but I'm having a hard time finding it.
In some sense the distribution is right-censored, but with a twist: for samples where I see no observation, it is unknown if $t>m$ or the event would never have occured at all. So the hope is the characterize the distribution without relying on the cases where the event is not observed.

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*Yes, it's fine. It's a perfectly ordinary thing to do. (That's not to say that there can't be some difficulties of implementation in some cases, but the basic idea is pretty standard.)


*If you are completely unaware of the existence of values outside some bounds, the distribution is truncated. This one is a truncated exponential, specifically right-truncated exponential.
https://en.wikipedia.org/wiki/Truncated_distribution
