# Maxwell Speed Distribution - Averages

Averages

There are three candidates for what is called the "average" value of the speed of the Maxwell speed distribution.

Firstly, by finding the maximum of the MSD (by differentiating, setting the derivative equal to zero and solving for the speed), we can determine the most probable speed. Calling this vmp, we find that:

Second, we can find the mean value of v from the MSD. Calling this :

Third and finally, we can find the root mean square of the speed by finding the expected value of v2. (Alternatively, and much simpler, we can solve it by using the equipartition theorem.) Calling this vrms:

Notice that

These are three different ways of defining the average velocity, and they are not numerically the same. It is important to be clear about which quantity is being used.

Probability distributions
Discrete univariate with finite support
• Benford
• Bernoulli
• Beta-binomial
• binomial
• categorical
• hypergeometric
• Poisson binomial
• discrete uniform
• Zipf
• Zipf-Mandelbrot
Discrete univariate with infinite support
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Continuous univariate supported on a semi-infinite interval, usually [0,∞)
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• Rayleigh
• relativistic Breit–Wigner
• Rice
• Rosin–Rammler
• shifted Gompertz
• truncated normal
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• Weibull
• Wilks' lambda
Continuous univariate supported on the whole real line (−∞, ∞)
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• Voigt
Continuous univariate with support whose type varies
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Mixed continuous-discrete univariate distributions
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Multivariate (joint)
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multinomial
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negative multinomial
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multivariate Student
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matrix t
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univariate von Mises
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bivariate von Mises
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Families
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• wrapped