# Thread: Expected value of Maxwell-Boltzmann distribution

1. ## 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)

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

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

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

would that do the trick?

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 $\displaystyle 0 \to v_0$, thus giving me a function I could solve for the integration limit $\displaystyle v_0$. Then I could simply integrate the original pdf with the derived integration limit to find the fraction of particles that have a $\displaystyle \bar{v}$ 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. ^_^