A family of pdfs or pmfs is called an exponential family if it can be expressed as \begin{equation} \label{eq:exp-fam} f(x\vert\boldsymbol{\theta}) = h(x)c(\boldsymbol{\theta})\exp{\left(\sum_{i=1}^{k}w_i(\boldsymbol{\theta})t_i(x)\right)}. \end{equation}
Theorem 3.3.1
If $X$ is a random variable with pdf or pmf of the form \ref{eq:exp-fam}, then
\begin{equation}
\label{eq:th1-e}
\E\left(\sum_{i=1}^{k} \frac{\partial w_i(\boldsymbol{\theta})}{\partial\theta_j} t_i(X)\right) = -\frac{\partial}{\partial\theta_j}\log c(\boldsymbol{\theta})
\end{equation}
\begin{equation}
\label{eq:th1-var}
Var\left(\sum_{i=1}^{k} \frac{\partial w_i(\boldsymbol{\theta})}{\partial\theta_j} t_i(X)\right) = -\frac{\partial^2}{\partial\theta_j^2}\log c(\boldsymbol{\theta})-\E\left(\sum_{i=1}^{k}\frac{\partial^2 w_i(\boldsymbol{\theta})}{\partial\theta_j^2}t_i(X)\right).
\end{equation}
The indicator function of a set $A$, most often denoted by $I_A(x),$ is the function \[I_A(x)= \begin{cases} 1 & x\in A \\ 0 & x \notin A. \end{cases} \]
A curved exponential family is a family of densities of the form \ref{eq:exp-fam} for which the dimension of the vector $\boldsymbol{\theta}$ is equal to $d<k.$ If $d=k$, the family of a full exponential family.