The average and variance of distributions (thermodynamics)

[pitch-black]

(Note: I'm not sure about international notations or terms, but I hope everything is comprehensible)

Next Monday I will pass my exam in theoretical physics about thermodynamics.
However, there's still one thing that I couldn't find explicitly described in my lecture notes or any additional literature.

It's the average and variance of distributions. All I found was the formulas, but no further explications.

Average:
<x> = integral (x * f(x)) dx

Variance:
< (x - <x>)^2 > = integral (x * (x - <x>)^2) dx

f(x) is in this case the Maxwell-Boltzmann distribution (such as f(x) = a * exp(-b*x^2) ).

What I don't know is what interval do I have to choose?
I thought about the whole set of real numbers, so from negative infinity to positive. However doing so, I don't get a sensible result, it's zero.
I also have thought about [0; infinity] or [0; x], but I actually have no idea.

Is the end result a term (including x) or a constant?


Any hints are highly appreciated.

 

[mathman]

(Note: I'm not sure about international notations or terms, but I hope everything is comprehensible)

Next Monday I will pass my exam in theoretical physics about thermodynamics.
However, there's still one thing that I couldn't find explicitly described in my lecture notes or any additional literature.

It's the average and variance of distributions. All I found was the formulas, but no further explications.

Average:
<x> = integral (x * f(x)) dx

Variance:
< (x - <x>)^2 > = integral (x * (x - <x>)^2) dx

f(x) is in this case the Maxwell-Boltzmann distribution (such as f(x) = a * exp(-b*x^2) ).

What I don't know is what interval do I have to choose?
I thought about the whole set of real numbers, so from negative infinity to positive. However doing so, I don't get a sensible result, it's zero.
I also have thought about [0; infinity] or [0; x], but I actually have no idea.

Is the end result a term (including x) or a constant?


Any hints are highly appreciated.

For the f(x) you described, the mean is 0.

Your variance formula has an error. It should read:

< (x - <x>)^2 > = integral (f(x) * (x - <x>)^2) dx
The variance = 1/2b, a is needed so that the integral of f(x)=1.

The integral is over the entire real line.

 

[pitch-black]

Thank you!
That helped me a lot!