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To obtain A I would integrate the pdf over the all possible values of v. The question is, should the limits be (-infinity,infinity) or [0,infinity) ? It seems that only by choosing the former can I get the correct normalization, but if in essence this is derived from an energy distribution , why doesn't it run from 0 to infinity?

If I try to work out <v> or <v^2> of this distribution, should I use (-infinity,infinity) or [0,infinity) ? because I always use the latter when working with the standard 3D maxwell boltzmann distribution (normalization, <v>, <v^2>)