About maxwell-boltzmann distribution

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The discussion centers on the Maxwell-Boltzmann distribution and its application in calculating the average velocity of gas molecules. The formula presented, average velocity = ∫ vf(v) dv, is clarified by a participant who explains that f(v) represents a probability distribution. This means that the normalization has already been accounted for, eliminating the need to divide by the total number of molecules (N) in the calculation of average velocity.

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Base on maxwell-boltzmann distribution,we can calculate the average velocity of gas molecule .
i read a book and it proves like that
average velocity=\int vf(v) dv
I don't understand why we don't have to divide it by N because we should be calculating the average,but not the total.
Would anyone tell me where i went wrong
 
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f(v) is a probability distribution, not the total number so you have already divided by N when calculating the probability distribution.
 
by multiplying each bin of the velocities, you already normalize the velocity distribution
 

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