(Note: I'm not sure about international notations or terms, but I hope everything is comprehensible) Next Monday I will pass my exam in theoretical physics about thermodynamics. However, there's still one thing that I couldn't find explicitly described in my lecture notes or any additional literature. It's the average and variance of distributions. All I found was the formulas, but no further explications. Average: <x> = integral (x * f(x)) dx Variance: < (x - <x>)^2 > = integral (x * (x - <x>)^2) dx f(x) is in this case the Maxwell-Boltzmann distribution (such as f(x) = a * exp(-b*x^2) ). What I don't know is what interval do I have to choose? I thought about the whole set of real numbers, so from negative infinity to positive. However doing so, I don't get a sensible result, it's zero. I also have thought about [0; infinity] or [0; x], but I actually have no idea. Is the end result a term (including x) or a constant? Any hints are highly appreciated.