Background Info:
I am looking for a more general form for the exponential family of distributions. Let's start by considering the normal distribution:
$$p(y|\mu,\sigma^2)= \frac{1}{\sqrt{2\pi} \sigma}\exp(\{-\frac{1}{2\sigma^2}(y-\mu)^2\})$$.
Question:
Demonstrate that the normal distribution (without considering the variance equal to 1) is an exponential distribution. Specifically, identify the terms $b(y)$, $\eta$, $T(y)$, and $a(\eta)$ in the exponential distribution probability function, which is given by:
$$p(y;\eta)=b(y)\exp\{\eta^\top T(y) - a(\eta)\}$$.
Since $\sigma^2$ is a variable here, $\eta$ and $T(y)$ will be two-dimensional vectors. Assume $\eta = [\eta_1 \ \eta_2]^\top$ for consistent notation. Also, express $a(\eta)$ in terms of $\eta_1$ and $\eta_2$.