Determine traffic speed distribution while driving I assume that the vehicle speed of cars on a highway is normally distributed around the posted speed limit. I could verify this by sitting by the road with a radar gun and measure vehicle speed for a large number of vehicles passing by.
How could I do this while driving at exactly the speed limit?
Assuming that I have a radar gun in my car, I would measure a positive speed for cars passing me and a negative speed for cars I pass. If I then add my own speed to the observed speed and I should get the same distribution as if I were sitting by the side of the road.
But I feel that I'm missing something: if I keep to the speed limit, then during any period of time, I observe a lot of cars that are either a lot slower than me or a lot faster than me. I'll see a few cars that are a bit slower or a bit faster. However, I'll see (almost) no cars that drive the speed limit (other than the ones in my direct vicinity), because I will never be able to observe them. As a result, I will get a lot of observations in the tail ends of the distribution and almost none near the mean (almost as if sampling from the inverse distribution...?).

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*Can I do a test on my observations to assert whether they fit a Gaussian distribution in the first place?

*If yes, what operation or transformation do I need to apply to my observations to get a Gaussian distribution, if possible?

*What if I can't actually measure speed, but only classify vehicles in bins ("a bit faster/slower, a lot faster/slower, ridiculously fast/slow")?


This crossed my mind the other day while driving and I had the feeling a lot more cars were passing me than I was passing (which is probably just a bias, but nevertheless it triggered this question). I have no background in statistics other than the basics from first year university, so this may be a solved problem and I just don't know what terminology to use to find the solution.

I'm going to address the idealized situation where we have a stationary density of cars per unit length along the road, per unit speed, $\rho(v)$. This distribution is time and space invariant.  For concreteness I'm going to write $\rho(v) = \rho(v ; \mu, \sigma^2)$ to express the fact that the distribution has a location $\mu$ and scale $\sigma$ parameter, though nothing in what follows requires the distribution to Gaussian.
We're going to be considering frames of reference moving with speed $v_0$.  In that case we have $\rho(u=v-v_0)du = \rho(u; \mu-v_0, \sigma^2)du$ (this is just variable substitution, and, physically, is a manifestation of Galilean invariance).
The rate at which cars will pass a fixed reference point in the moving reference frame will be given by $I(u)du  = u \rho(u) du= (v-v_0)\rho(v-v_0; \mu-v_0, \sigma^2)dv$.  The reasoning behind this goes like this:  How many cars moving at (relative) speed $u$ pass a reference point $x_0$ in the (moving) reference frame between times $t$ and $t+\delta t$?  All of the cars with (relative) speed $u$ at locations between $x-u(t+\delta t)$ and $x-u t$. The density (per unit speed) is $\rho(u)*u*\delta t$, so the rate per unit time is $u\rho(u)du$
So in the end, the rate (cars/unit time) you'll see cars moving with relative speed in the (infinitesimal) interval $[u, u+du]$  when moving at speed $v_0$ relative to the road is
$I(u; v_0)du = u \rho(u; \mu-v_0, \sigma^2)du$.
