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Math Help - Expected value of Maxwell-Boltzmann distribution

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    Expected value of Maxwell-Boltzmann distribution

    Hi everyone,

    I'm solving a problem that requires me to find the average velocity of a portion of the Maxwell-Boltzmann velocity distribution. I know that the expected value for a probability distribution f(x)

    \bar{x} = \int^\infty_{-\infty} x\cdot f(x)\,dx

    But how would you go about computing the average value of part of the distribution? Say I need to find the average velocity of some fraction of the particles on a Maxwell-Boltzmann distribution (f(v)) up to a certain. If I just change the integration limits, giving

    \bar{v} = \int^a_0 v\cdot f(v)\,dv

    would that do the trick?

    Thanks in advance,

    JT

    P.S. More accurately, the problem is asking me to calculate the fraction of particles on a certain M-B pdf that gives a pdf corresponding to a given temperature. I initially decided that, since I could find the average velocity corresponding to the target temperature, I could equate that with the above integral of the pdf at the initial temperature, with integration limits 0 \to v_0, thus giving me a function I could solve for the integration limit v_0. Then I could simply integrate the original pdf with the derived integration limit to find the fraction of particles that have a  \bar{v} that gives the target temperature. However, my problem is that I'm not sure if my assumption about the expected value formula is valid, hence the question. ^_^
    Last edited by Psilocybe; November 15th 2010 at 03:37 PM. Reason: additional info
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