In generalized linear model ,the distribution of response variables follow the exponential family distribution form :
$$ P(x\mid \eta)=h(x)\exp\big(\eta^Tt(x)-a(\eta)\big) $$
My question is, when discuss the expectation value of the exponential family ,why we always focus on $\mathbb{E}\big[t(x)\big]=\dfrac{\partial a(\eta)}{\partial\eta}$ ? Is there any relationship between $\mathbb{E}\big[x\big]$ and $\mathbb{E}\big[t(x)\big]$ ?