# Why Cauchy Distribution isn't Exponental Family?

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?

• I'm sorry, the probability density function for Cauchy Distribution with parameter $\lambda$ should be $p(x;\lambda)=\frac{\lambda}{\pi (\lambda^2 +x^2)}$ – Victor Chen Mar 29 '16 at 16:43
• You can use the "edit" button under the question. – zhoraster Mar 29 '16 at 16:47

Hint: If $X$ were from exponential family, it would have finite expectation (you may even express it in terms of $A(\theta)$).
• Can you explain why $X$ must have finite expactation? – Daniel Yefimov Dec 7 '17 at 13:10
• @Daniel Yefimov Because we assumed $X$ is from exponential family. Check Bartlett identity. – Zhanxiong Dec 7 '17 at 13:38