Might be a stupid question though...
we know for Exponential Family, we have the density is $$f(x|\theta)=h(x)\exp[\eta(\theta)\cdot T(x)-A(\theta)]$$ Is there a general form for the CDF function? I know if we specify a member from the exponential family, like normal, exponential, etc, the question would be simple. But is there a general form of the CDF?
Thanks~
Well, a general form is straight yielded by the expression of $f(x|\theta)$. Nothing more than that though, as there are cases where you can't have a CDF in terms of a closed form, for example the gaussian or the normal distribution, which yield non-common functions).
For continuous distributions belonging to the Exponential Family (for a random sample-distribution defined over the space $x \in \mathcal{X}$):
$$F_x(x|\theta) = \int_{x \in \mathcal{X}} f(x|\theta)\mathrm{d}x = \int_{x \in \mathcal{X}} f(x|\theta)\mathrm{d}x = \int_{x \in \mathcal{X}} \bigg\{h(x)\exp[\eta(\theta)\cdot T(x)-A(\theta)]\bigg\}\mathrm{d}x $$
For discrete distributions belonging to the Exponential Family (for a random sample $X_1, \dots, X_n$) :
$$F_x(x|\theta) = \sum_{i=1}^n f(x_i|\theta) = \sum_{i=1}^n \bigg\{h(x_i)\exp[\eta(\theta)\cdot T(x_i)-A(\theta)]\bigg\}$$