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I'm trying to write the chi-squared distribution in the form of: $$\exp\big\{(y\theta-b(\theta))/(\phi/w)-c(y,\phi)\big\}.$$

Using a shape-mean paramatization of the gamma distribution (i.e. taking shape parameter $\alpha$ and rate $\beta$, then re-parametrizing the $\beta$ parameter with $\mu=\alpha/\beta$, I was able to write the gamma distribution in this form, with: $$\theta=-1/\mu,\ b(\theta) = -\log(-\theta),\ \phi=1/\alpha,\ w=1,$$ and $$c(y,\phi) = \log(\Gamma(\alpha))-\alpha \log(\alpha)-(\alpha-1)\log(y).$$

As the chi-square is a special case of the gamma distribution with shape $\alpha=k/2$ and rate $\beta=1/2$, I initially tried using this. However I end up with $\theta=-1/k$ and $\phi=2/k$. In other words, the top and bottom of the first term in the exponential pdf form are literally just multiplied by $1/k$ in order to force a canonical parameter in there. This seems a bit weird to me - especially as if $\phi$ is a function of $\theta$, then $c(y,\phi)$ is a function of $\theta$ too, which it shouldn't be.

Other than that, I get a canonical parameter of $\theta=-1$. However, the $b()$ function is not meant to rely upon $\phi$, so there is no way of having $b'(\theta)=k$.

tldr: how do I go about expressing the chi-square pdf in the form of the exponential family of probability distribution?

Sorry if it's a bit hard to read! thanks!

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