Fourier analysis and prob. distributions?

[Zaphodx57x]

Ok, this might seem like either a really idiotic question or a really profound one.

Consider a probability distribution. I'm picturing a normal distribution, is it meaningful to be able to build up a final probability distribution from a set of narrower probability distributions?

Ok, that seems like it came out really poorly so i'll say a few of my thoughts. In quantum mechanics we use $$\Psi$$(r,t) to represent the wave function for very small particles. Then we square this to get |$$\Psi(r)|^2$$ which is the probability density. This, I believe would then give me a probability distribution. Which in a lot of physics examples is just some multiple of a sine wave. Now, it seems to me(being a novice at both probability and physics) that it may be possible to build up a probability distribution of this sort from several smaller probability distributions through simple interference plotting or Fourier analysis or the like.

However, I can't resolve to myself why this would be a meaningul thing to do. For instance, multiple probability distributions might imply multiple wave functions and hence multiple particles. And multiple particles would interact usually; thus changing the original wave functions and doing something funky.

Can anyone comment on this?

 

[Zaphodx57x]

I guess profundity is ruled out.

 

[Hurkyl]

I have no idea what you intend to do.


You do realize that we can do all sorts of arithmetic on random variables, right? For instance, we can add them, multiply them, square them, divide them, take their logarithm, etc...

 

[Data]

Now, it seems to me(being a novice at both probability and physics) that it may be possible to build up a probability distribution of this sort from several smaller probability distributions through simple interference plotting or Fourier analysis or the like.

However, I can't resolve to myself why this would be a meaningul thing to do. For instance, multiple probability distributions might imply multiple wave functions and hence multiple particles. And multiple particles would interact usually; thus changing the original wave functions and doing something funky.

Indeed. This is just the superposition of wavefunctions. An obvious example is the interference pattern observed in double-slit electron diffraction experiments.

 

[Zaphodx57x]

Thanks for the reply. I'll play with it a little and see what I can get out of it.

 

[Watts]

Probability Mixing

You can easily write a single distribution as the sum of two or more. An example is given below. It’s called probability mixing. I suppose you could do the same thing for the magnitude of the wave function.

exp(-pi*x^2)=(1/2)*exp(-pi*x^2)+(1/2)*exp(-pi*x^2)