I have a problem, where I have to show that the Cauchy distribution has zero mean. I'm in doubt about if I have to show, that the expectation does not exists or it is zero?
The function is given by $f(x)=\frac{1}{\pi}\frac{1}{x^2+1}$ with the interval $(-\infty,\infty)$. If I just have to explain that the mean is zero, I guess it is because of the symmetric interval around zero?
I've also shown that the integral does not exists, by taking the limits $[0,\infty)$, which leads to that the expectation becomes infinite. So by a theorem, I have that since the expectation is infinite, then the expectation does not exists. But I cannot really explain why I should take the limits from $[0,\infty)$?
Thanks for the help in advance.
This is simply wrong.
The expectation is defined only when the integral $$ \int f(x)|x|dx $$is finite, but here:
$$ \int f(x)|x|dx = 2\int _0^\infty \frac {xdx}{(1+x^2)\pi} = \infty $$ as $\frac {x}{1+x^2} \sim_\infty \frac1x$ and $\int \frac {dx} x= \infty$