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One thing that always confused me about the exponential distribution was why the probability distribution function or even the expected value is modeled as a function of $X$ instead of a variable like $T$ for time--note capitalization means a random variable. This is not meant to be a nit-picky notation question, but really just trying to understand if I am missing something about the distribution and why $X$ is used.

The exponential is used for waiting time or inter-arrival times in general.

If you look at the Wikipedia article for example, the pdf is specified as: $$ f(x;\lambda) = \lambda e^{-\lambda x} $$

The exponential distribution models the probability that an event happens within some continuous interval $[0, t]$, given an intensity parameter $\lambda$. Now other distributions such as the poisson distribution, explicitly reference time as the variable of interest: $P(k, t; \lambda) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$

Using $x$, the notation for the pdf of the exponential becomes something awkward like $ P(X \leq T; \lambda)$ or equivalently $P(x \in [0, T])$. Wouldn't it seem more intuitive to specify the pdf as $P(T \leq t; \lambda)$, where the random variable is specified as $T$?

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Wouldn't it seem more intuitive to specify the pdf as $P(T≤t;λ)$, where the random variable is specified as T?

The simple answer is no. "Intuitive" is context dependent term. For me, $X$ for exponential r.v. is much more intuitive than $T$, as I'm used to the standard notations of capital $X$ and $Y$ for random variables. Whatever they designate of applied to. You are concentrated much on the application, however, both exponential and Poisson distributions exist independently of their application and the very notion of time. E.g., Poisson is frequently used to count something, however you don't have to involve the time in this. You can count events per square meter, or per any other time independent measure-unit, or per subject, etc.

Regarding the exponential r.v., note that exponential distribution is a special case of Gamma distribution with $\alpha = 1$, and pace $\lambda$. Where Gamma is not associated that much with time-to-event modelling like the exponential distribution, so it is natural to denote Gamma by $X$, and thus it will be weird switch to $T$ for a special case of $\alpha=1$.

However, note that notations are generally context dependent. Such that, in the context of survival analysis, time-to-event is usually denoted by $T$, where $T$ can be distributed exponentially. The same holds for queuing theory, the waiting times are also usually denoted by $T$. That is, where the context is time, the $T$ notation is adopted, however for general introduction, there is no special reason to use $T$ and not the common notation of $X$.

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    $\begingroup$ yes, this make sense. I get what you mean about the difference between the model versus its application. Perhaps my confusion was due to the way that the book presented it--they included the $t$ variable when defining the Poisson process, but then took the $t$ away when talking exponentials. But thanks for reinforcing the intuition here, even if the notation was a bit confusing to me. $\endgroup$ Commented Jan 29, 2021 at 17:46
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I guess so, but actually we use the letter $X$, $Y$ and so on because it is a standard idea to use such letters. Of course "standard" depends on the context. In probability theory or in general in statistics, we mostly use $t$ when talking about $t$-student distribution.

In any case you are entitled to use the letter you like the most, as long as you specify it: nobody can tell you anything if you write: $$T\sim Exp(\lambda)$$ Notice moreover that what really matters is how you write the variable you referr to: Exp will always denote an exponential, $\textit{Po}$ a Poisson, $\mathcal{N}$ a normal and so on.

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