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We know that the density function of standard Cauchy Distribution is $p(x)=\frac{\lambda}{\pi (\lambda^2+x^2)}$.

It seems that $p(x)$ can't be written as the form of Exponential Family $$f_{X}(x;\theta)=h(x)\exp\Big(\sum^{s}_{i=1}{\eta_i (\theta)T_i (x)-A(\theta)}\Big)$$

But how to prove it strictly? or how to show that Cauchy Distribution doesn't belong to the Exponential Family?

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  • $\begingroup$ I'm sorry, the probability density function for Cauchy Distribution with parameter $\lambda$ should be $p(x;\lambda)=\frac{\lambda}{\pi (\lambda^2 +x^2)}$ $\endgroup$ – Victor Chen Mar 29 '16 at 16:43
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    $\begingroup$ You can use the "edit" button under the question. $\endgroup$ – zhoraster Mar 29 '16 at 16:47
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Hint: If $X$ were from exponential family, it would have finite expectation (you may even express it in terms of $A(\theta)$).

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    $\begingroup$ Can you explain why $X$ must have finite expactation? $\endgroup$ – Daniel Yefimov Dec 7 '17 at 13:10
  • $\begingroup$ @Daniel Yefimov Because we assumed $X$ is from exponential family. Check Bartlett identity. $\endgroup$ – Zhanxiong Dec 7 '17 at 13:38
  • $\begingroup$ The sufficient statistic must have finite expectation. $\endgroup$ – Sextus Empiricus Jun 12 '20 at 21:28

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