I haven't taken any classes in Statistical Mechanics, but in studying Structure of Matter I found some ideas I'm not very familiar with, related with the average value of energy ($E$). Given a $p(E)$ probability density, the average energy is:
\begin{equation} \langle E\rangle=\frac{\int\limits_0^\infty Ep(E)dE}{\int\limits_0^\infty p(E)dE} \tag{1} \end{equation}
Now, in two different cases the average energy is calculated using either the Boltzmann distribution (the average energy per normal mode, when deriving Rayleigh Jeans) \begin{equation} p(E)\propto e^{-\frac{E}{kT}} \end{equation} or the Maxwell-Boltzmann(*) distribution (when deriving $\beta=\frac{1}{kT}$.). \begin{equation} p(E)\propto\sqrt{E}e^{-\frac{E}{kT}} \end{equation} The first case yields:
\begin{equation} E=kT \tag{2} \end{equation}
While the second case yields:
\begin{equation} E=\frac{3}{2}kT \tag{3} \end{equation}
That is familiar from kinetic theory.
I suppose I am calculating average energy in two different situations.
Can you provide some physical (mathematically is quite clear, the distributions being different) insight on why this results are different in $(2)$ and $(3)$?
(*)
Actually it is not directly stated. It "builds" average energy starting from
\begin{equation}
\langle E\rangle=\frac{\int\limits_0^\infty Edn}{\int\limits_0^\infty dn}
\end{equation}
and then $dn$ is expressed in term of $dE$ and there is this $\sqrt{E}$ factor, so I assumed the book was implicitly using Maxwell Boltzmann.
(Brehm, Introdution to the structure of matter. Chapter 2, section 3, example 2)