I'm having trouble putting words on a concept.
I'm comparing some Monte-Carlo data with uniformly random distributed data.
The random data is not uniform in a normal rectangular coordinate system, but it is uniform over pseudorapidity.
Pseudorapidity is a coordinate system used in accelerator physics, described as follows.
$$\eta = -\ln(\tan(\frac{\theta}{2}))$$ where $$0 \lt \theta \leq \frac{\pi}{2}$$
I used the inverse cdf sampling function to create random samples that fit this function. No biggie. But I don't know how to describe the resulting data to my colleagues. I've settled for now on "random samples of uniform density over pseudorapidity", but it leaves a weird taste in my mouth.
Does anyone know anything better to call it?
For $\theta\in(0,\pi/2]$ we have $\eta\in[0,\infty),$ so the distribution of $\eta$ cannot be uniform. (There exists no uniform distribution over an unbounded interval.) Thus, the phrase "uniform over pseudorapidity" is nonsensical in this context. If you sampled angles $\theta_i$ uniformly at random over $(0,\pi/2]$ and simply computed the corresponding $\eta_i = -\ln(\tan(\frac{\theta_i}{2}))$, then these $\eta_i$ would constitute a (non-uniform) random sample of pseudorapidities induced by that uniform distribution of angles.