Out Of Sight, Still In Mind

Is there an approximation to the expression for the probability of a particle being in energy partition $E_j$ ,

$p(E_j)=\cfrac{(\eta E_j)^{j/2}e^{-\sqrt{\eta E_j}}}{j!}$?

we can rewrite,

$p(E_j)=e^{\large{jln\left\{\sqrt{\eta E_j}\right\}-\sqrt{\eta E_j}}}.\cfrac{1}{\Gamma(j+1)}$

But where's $T$, the temperature of the system?

$T$ is in $E_j$.  In this equation there is only one expression for energy.  The energy of a particle is not divided into $E$ and $T$, where $T$ is part of $E$.

Is this still useful?

$T$ is often fixed at 298.15 K in calculations involving Boltzmann or Fermi-Dirac Distributions.

How does $E$ related to $T$ as we measure it?  Is $T\propto E$?