Why Cauchy Distribution isn't Exponental Family?

Asked Mar 29 '16 at 16:39

Active May 09 '25 at 04:13

Viewed 4,347 times

$$p(x)=\frac{\lambda}{\pi (\lambda^2+x^2)}$$

It seems that the density function of standard Cauchy Distribution above can't be written in 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 that Cauchy Distribution doesn't belong to the Exponential Family?

edited May 09 '25 at 04:13

Starlight

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asked Mar 29 '16 at 16:39

Victor Chen

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
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You can use the "edit" button under the question. – zhoraster Mar 29 '16 at 16:47

Why Cauchy Distribution isn't Exponental Family?

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