Determine the natural parameter space of the exponential family of distribution of dimension one with $\chi=1, T=x,h=e^{-x^2}$. And $h(x)=e^{-|x|}$.
Work:
The natural parameter space is the set of $\theta$ such that the integral in $A(\theta)=log\int h(x)e^{\theta T(x) }dx$ is finite. Here,
$$\begin{split}\int_{-\infty}^{\infty}h(x)e^{\theta T(x)}&=\int e^{-x^2}e^{\theta x}dx\\ &=\int e^{-x^2+\theta x}dx\end{split}$$
which I do not know how to integrate. Is this...right?
