From what I understand, the exponential family is defined as

$$f(y;\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)}+c(y,\phi)\right) $$

I've read (but not seen shown anywhere), that the t distribution is not a member of the exponential family. But I don't understand why.

For instance, suppose I set $\theta = 0$, $b(\theta)=0$, and set $$c(y,\phi) = \ln\left(\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}\right)$$,

wouldn't then the t distribution then appear because

you would have

$$\exp\left(0+\ln\left(\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}\right)\right)=\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}$$

which is the t-distribution.

Why doesn't this work?


Ideally, I'd like to know why also my logic above is wrong (which I'm certain it is). So if you can fit that into your answer, that would be great.

  • 3
    $\begingroup$ Because the random variable and the parameter cannot separate as a product. $\endgroup$
    – Xi'an
    Jan 12, 2020 at 20:19
  • 2
    $\begingroup$ Your characterization of an exponential family is meaningless because every density function $f$ can be written in such a form: simply set $\theta=0=b(\theta)$ and $c(y;\phi)=\log f(y;\phi).$ $\endgroup$
    – whuber
    Jan 12, 2020 at 20:38
  • 1
    $\begingroup$ Your first equation is the definition of an exponential dispersion model, not the exponential family, and note that $a()$ is required to be positive. $\endgroup$ Jan 13, 2020 at 4:34
  • 1
    $\begingroup$ If $\phi$ is constant then it becomes a linear exponential family, but if $\theta$ is constant (as in your calculation) then it isn't anything at all. $\endgroup$ Jan 13, 2020 at 4:40

1 Answer 1


No, the t distribution is not an exponential family. Exponential family distributions do have existing moment generating functions, and the t distribution do not. See also Why doesn't the exponential family include all distributions?


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