From what I already know, to calculate the expectation value/average from a probability distribution, you use the formula:

$$ \langle x \rangle \ = \int_{-\infty}^{\infty} x f(x) \,\mathrm{d}x \tag{1}$$

where $f(x)$ is the probability distribution of some variable $x$.

However, in my lecture notes, the average kinetic energy $ \langle E \rangle $ of gas particles in thermal equilibrium using the Maxwell Boltzmann distribution $f(u)$ has been given as:

$$ \langle E \rangle \ = \frac{\int_{-\infty}^{\infty} \frac{1}{2}mu^2 f(u) \,\mathrm{d}u}{\int_{-\infty}^{\infty} f(u) \,\mathrm{d}u} \tag{2}$$

where $u$ is the velocity.

I don't understand why there is a division by $\int_{-\infty}^{\infty} f(u) \,\mathrm{d}u$ in Equation (2), unlike what I've previously learned as in Equation (1), and I can't find the same form elsewhere.

If anyone has seen this and would be able to explain it to me, I would really appreciate it!

  • 3
    $\begingroup$ The eq. (1) implies that $f(x)$ is true probability measure, i.e. the integral over all possible values of random variable is unity. The eq. (2) contains division by normalization constant and it is two-step procedure: 1) we compute normalization constant, 2) we compute average with true (normalized) probability measure $\endgroup$ Commented Dec 31, 2023 at 13:34
  • $\begingroup$ @ArtemAlexandrov Ah okay that makes sense, thanks so much! And thank you for answering so quickly as well! $\endgroup$
    – user374355
    Commented Dec 31, 2023 at 14:07

1 Answer 1


The formula given for $\langle E\rangle $is useful for probability distribution that is not normalized. The formulas can be made to look like each other by defining the normalization constant $N$ as

$$ \int_{-\infty}^\infty f(u) du =N $$ Which we can since this integral is just a number and then we can write the normalized $f(u)/N = g(u)$ so then the formula provided in (1) applies for $ g(u)$.

$$\langle E\rangle = \int \frac{1}{2} mu^2 g(u) du$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.