[QUE/SM-04022] SM-PROBLEM

Consider a system having the probability of being in the state with label $i$ as $p_i$. Let an extensive variable $X$ take the value $X_i$ in the $i$th stat. We have $\sum_{i=1}^Np_i\,=\,1$ and the average value of $X$ for the system is fixed at $\overline{X}$ Show that for the system to be in equilibrium
$$ p_i\,=\,\frac{e^{-KX_i}}{\sum_{j=1}^Ne^{-Kx_j}}$$
$K$ is an undetermined multiplier. Find the entropy of the system in terms of the Boltzmann constant, $k$, $\overline{X}$ and $Z\,=\,\sum_{j=1}^Ne^{-Kx_j}$