Given two successive Gaussian events,
$X_1 \sim N(\mu,\sigma^2),\qquad X_2\sim N(\mu,\sigma^2)\qquad$ with $X_2 = X_1+\epsilon$,$\quad\epsilon\in {\rm I\!R},\quad\epsilon>0$
how can I derive the distribution function of the waiting time, $\epsilon$, between them?
$X_2 - X_1 = \epsilon \sim ~?$
In other words, if the time at which a certain event is detected follows a Gaussian distribution, what is, or how can I derive, the distribution of the time interval between two successive events?
[Edit]
My original question was much less detailed than I originally thought, so I will try to give a better example.
Suppose I have a time series with random peaks. If I look at the distribution of those peaks in time I will see a Gaussian distribution of times.
Now I want to look at the distribution of the time interval between peaks (or waiting time). I get this distribution that looks like an exponential distribution. But I know it isn't a true exponential distribution. I want to know which distribution it is or how to derive it.
