The exponential family of probability density functions take the form
$f(y,\theta, \phi) = \exp \bigg\{ \frac{y\theta - b(\theta)}{a(\phi)} + c(y, \phi) \bigg\}$. I know that $$ E(\frac{\partial \text{log-likelihood of }f}{\partial \theta}) = 0$$.
I've tried to prove it, but I don't understand how one could take the expected value of a partial derivative. How would I prove this?
Edit:
In more detail, the expected value is given by
$$ \int \frac{\partial \log L}{\partial \theta} g $$
integrated over the domain of $\frac{\partial \log L}{\partial \theta}$ and where $g$ is the density of $\frac{\partial \log L}{\partial \theta}$. I'm not sure how to find $g$ and solve this integral.
