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Does the moment generating function of a Cauchy distribution given below exist? How do I show that the mean of the Cauchy does not exist? Please help. I am stuck on how to go about it. $$ \int_{-\infty}^{\infty} e^{tx} \frac{1}{\pi(1+(x-\theta)^2)} \mathrm dx $$

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    $\begingroup$ The MGF does not exist because this integral is not absolutely convergent. Same goes for the mean and any higher moments. See the theoretical answers in this thread. Or by showing that the mean does not exist, conclude that the MGF fails to exist. $\endgroup$ – StubbornAtom Oct 6 '18 at 20:53
  • $\begingroup$ On the other hand, the characteristic function does exist and is $E\left[\exp(itX)\right]=\exp(i \theta t - |t|)$ $\endgroup$ – Henry Oct 7 '18 at 8:39
  • $\begingroup$ @Henry, I am stuck on how to show that. Can you please explain more? $\endgroup$ – Lady Oct 7 '18 at 8:40
  • $\begingroup$ This and that have some possible approaches $\endgroup$ – Henry Oct 7 '18 at 8:45

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