Assigning velocity to particles using probability distribution

Create the cumulative distribution function, and map a set of linear random numbers [0,1) to velocities by using that.

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Create the cumulative distribution function,

I searched the internet for one but I can't find it.

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map a set of linear random numbers [0,1) to velocities by using that.

Can you elaborate a bit more on this because I'm not sure what you mean.

1. Generate a (realization of a) uniform random variable x on the interval [0,1].
2. Compute y = F^-1(x).
3. y is a (realization of a) random variable distributed according to F.

This is the Inversion method, and it's usually the best thing to do, though occasionally you can use rejection sampling instead because it's easy.

Now, I realized that you're interested in vector random variables. What Bob Janova and I described works for scalar random variables. How can we generalize?

Well, in your case, we don't have to. :-) Your problem leads to some drastic simplifications actually (I needed to brush up on my physics to realize this). First, let me write the Maxwell distribution over velocities (from Wikipedia):

Notice that this can be factored (as Wikipedia points out), written as the product of independent distributions in vx, vy, and vz. Hence, to generate v=(vx,vy,vz), you can just generate vx, vy, and vz independently according to the pdf,

f(vi) = [m/(2 pi k T)]^1/2 exp[-(m/(2 k T)) vi²]

which is a normal distribution; how to sample it (generate values distributed according to it) is described here.

More generally (in other problems), to sample a vector-valued random variable (z₁,...,z_n) what you can do is,
1. Generate z₁ according to p(z₁)
2. Generate z₂ according to p(z₂ | z₁)
...
3. Generate z_n according to p(z_n | z₁, ... z_n-1)

Thanks for your help. Unfortunately, I won't have time to try this out until next week. I'll post my results then.