The relationship between mean and variance in the context of system energy and the partition function I'm looking at a specific derivation on wikipedia relevant to statistical mechanics and I don't understand a step.
$$ Z = \sum_s{e^{-\beta E_s}} $$
$Z$ (the partition function) encodes information about a physical system. $E_s$ is the energy of a particular system state. $Z$ is found by summing over all possible system states.
The expected value of $E$ is found to be:
$$ \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} $$
Why is the variance of $E$ simply defined as:
$$ \langle(E - \langle E\rangle)^2\rangle = \frac{\partial^2 \ln Z}{\partial \beta^2} $$
just a partial derivative of the mean. 
What about this problem links the variance and mean in this way?
 A: The answer is valid for the partition sum $Z$ (which is closely related to the moment generating function). The reason is the special structure of the partition sum $$Z = \sum_s e^{-\beta E_s}.$$
The system is characterized
with probability $$P_s=\frac{e^{-\beta E_s}}{Z}$$  that a state $s$ with energy $E_s$ is attained.
Given this definition it is easy to see that
$$-\partial_\beta \ln Z = -\frac{\partial_\beta  Z}{Z} 
 = \sum_s E_s  \frac{e^{-\beta E_s}}{Z}= \sum_s P_s E_s =\langle E \rangle  .$$
Similarly, one can easily convince oneself that 
$$
\begin{align*}
\partial_\beta^2 \ln Z &= -\partial_\beta \left[ \sum_s E_s  \frac{e^{-\beta E_s}}{Z} \right]
=\sum_s E_s^2  \frac{e^{-\beta E_s}}{Z} - \left[ \sum_s E_s  \frac{e^{-\beta E_s}}{Z}\right] \left[\sum_{s'} E_{s'}  \frac{e^{-\beta E_{s'}}}{Z}\right]\\
&= \langle E^2\rangle -\langle E\rangle^2 = \langle (E- \langle E\rangle)^2\rangle,
\end{align*}$$
i.e., the variance is given by the second derivative of $\ln Z$.
A: Have you seen the link of the definition of variance or expected value in wiki, here?
A: The variance of a random variable $X$ is always defined as $<(X - <X>)^2>$; this is the expected square of the difference between the expected and actual values.
