Now that we have a probability function, it is important to determine its mean and variance. It should be obvious that when doing so using the Cauchy probability function, problems quickly arise. Indeed,
\begin{equation*}
\int_{-\infty}^{\infty} x \frac{1}{1+x^2} dx = (1/2) ( \ln( | \infty |) - \ln( | -\infty |)
\end{equation*}
which is problematic. Further, even assuming that the distribution is symmetrical and therefore has a mean of 0, for the variance
\begin{equation*}
\int_{-\infty}^{\infty} x^2 \frac{1}{1+x^2} dx
\end{equation*}
and note that the integrand does not converge to 0 at the endpoints and therefore the integral is automatically considered divergent. Thus it is reasonable to note that the Cauchy distribution has no variance.