Probability distribution with unbounded expectation $\frac{1}{\pi(1+x^2)}$ is a valid probability distribution - it integrates to $1.$ 
If I take its expectation, $\int_{-\infty}^\infty \frac x{\pi(1+x^2)} dx$, I get an unbounded value.
However, the distribution is symmetrical around zero, so intuitively the expectation should be zero.
How I do I reconcile those two facts? What is the expectation of the distribution?

The mean is not defined. In reality you could compute the mean by computing the integral from -a to a and then taking $a\to \infty$, that is $\lim_{a\to \infty}\int_{-a}^a xf(x)dx = 0$ but you can also do: $\lim_{a\to 0} \int_{-2a}^a xf(x)dx \neq 0$ so in reality you can have any value. Empirically this means that if you generate random numbers from this distribution the empirical mean will not stabilize.
See http://en.wikipedia.org/wiki/Cauchy_distribution#Mean
