I'm interesting in finding the exponential family density associated with the function $$p_\theta(x) = \exp(\theta^T x - A(\theta)) \text{ with } x \in \mathbb{R}^n_+.$$
In this case, I have $\log A(\theta) = \int \exp(\theta^T x)\,dx$ where the integral is over the domain of $x$ (since $-\log A(\theta)$ is a normalizing function for the density)
I am trying to evaluate the integral to determine which exponential family is associated with the distribution. My exploration here took me to Gaussian integrals, namely trying to use the real multidimensional Gaussian integral (from here) but this doesn't work since the determinant in the resulting function is zero (and thus the integral would be zero). Is it possible to find a closed form solution to this multivariable integral?
Am I even going down the wrong route by trying to evaluate the integral at all? Based on the non-normalized probability, this appears to represent the join distribution of independent exponential random variables, but that is merely a hunch.