Writing exponential family in canonical form

Question

I have the following pdf with support $x>0$:

$$f_{\mu}(x)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{(x-\mu)^2}{2\mu^2x}\right)$$

This belongs to the exponential family, and I write this in the following form

$$f_{\mu}(x)=h(x)\space\textrm{exp}\left(\spaceη(\mu)T(x)\space-B(\mu)\right)$$

What I have thus far is by expanding the pdf is

$$h(x)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{1}{2x}\right), η(\mu)=-\frac{1}{2\mu^2}, T(x)=x \space\textrm{and}\space B(\mu)=\frac{1}{\mu}$$

Something is wrong here, because when I write this in canonical form, i.e.

$$f_{\theta}(x)=h(x)\space\textrm{exp}\big(\theta x-b(\theta)\big)$$

I get the following

$$\theta=-\frac{1}{2\mu^2} \space\textrm{and}\space b(\theta)=\sqrt{-\frac{1}{2\theta}}$$

I want to rewrite the pdf in terms of $\theta$. Without this, I am unable to find the expectation, variance, fisher and so on in terms of $\theta$.

Can someone please tell me where I went wrong?

Answer 1

Since $$f_{\mu}(x)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{(x-\mu)^2}{2\mu^2x}\right)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{x}{2\mu^2}+\frac{1}{\mu}-\frac{1}{2x}\right)$$ one gets that $$\eta(\mu)=\frac{-1}{2\mu^2}\qquad B(\mu)=-\frac{1}{\mu}$$ (hence a missing minus sign for $B$) and for the natural parameterisation $$\theta=\frac{-1}{2\mu^2}\qquad b(\theta)=-\sqrt{-2\theta}$$ (hence the inverse of the proposed $b$).