I am trying to compute the mean $\bar{E}$ of a Boltzmann distribution $$ p(\pmb{x}) = \frac{1}{Z} e^{-\beta E(\pmb{x})}, $$ knowing that there is a total of $N$ microstates $\pmb{x}$ and $[\alpha N]$ energy levels, and that each energy level has degeneracy ${N}\choose{E}$, $E \in \{1, 2, \ldots, [\alpha N]\}$, for some $\alpha$ is fixed in $[0, 1]$.
In the case $\alpha=1$, I was able to compute $\bar{E}$ analytically from the partition function: $$ Z = \sum_{\pmb{x}} e^{-\beta E(\pmb{x})} = \sum_{E=0}^{N} {{N}\choose{E}} e^{-\beta E} = (1 + e^{-\beta})^N $$ and so $$ \bar{E} = - \frac{\partial}{\partial \beta}\log Z = \frac{N}{1 + e^\beta}. $$
I am not able to compute $\bar{E}$ for a generic $\alpha \in [0, 1]$, though. In particular, it seems that there is no analytic expression for the truncated sum $\sum_{E=0}^{[\alpha N]} {{N}\choose{E}} e^{-\beta E}$ (see https://mathoverflow.net/questions/93744/estimating-a-partial-sum-of-weighted-binomial-coefficients?rq=1).
My final goal though is not to compute $Z$, but to study $\bar{E}$ as a function of $\beta$, for any fixed $\alpha$ in the unit interval. Ideally, I would like to have a way to compute $\beta$ given $\bar{E}$. Any ideas on how I could go about this? Approximations of $\beta$ given $\bar{E}$ would also be very useful. Thanks!
