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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$\begingroup$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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