I have a sequence of real numbers. I want to know whether this sequence looks like the traces in the standard representation of a random sequence of elements of $G_2$. (Here random is according to the Haar measure on the compact form.)

So I want to compare my sequence with the measure on $\mathbb R$ that is the pushforward of the Haar measure along the trace map. Thus I want some description of this measure.

It seems difficult to find an explicit formula for this measure. However, I'd still like a way to estimate it.

For $U_n, SU_n$, $USP_n$, and $O_n$, this paper describes how to generate a random element, and hence how to estimate the Haar measure using Monte Carlo.

One could also compute the moments of the distribution using representation theory, and try to estimate the measure from that, but this seems like a bad idea.